Straight from the DMG Errata:

Attunement (p. 136)

The first paragraph ends with a new sentence: “If the prerequisite is to be a spellcaster, a creature qualifies if it can cast at least one spell using its traits or features, not using a magic item or the like.”

emphasis added

Therefore, for the purpose of attunement:

If a creature is able to cast a spell by virtue of their own ability (not from an item), whether that be race, class, or feature, you are a spellcaster.

I've just finished running this module, with great success, so I can address a few of your concerns..

First, I challenge the premise that Dead in Thay needs to be a "fast-paced assault". Your players signed up for this adventure not because it was a "fast-paced assault", but because it was a behemoth of a dungeon. They knew full well it would be a long slog. So give your players what they expect, which is a vast and varied dungeon loaded with combat and interesting encounters. Don't stress too much about the pace not quite being fast enough.

But it can still be very easy for Dead in Thay to become drawn out and long. Some of the encounters can get repetitive ("Oh, another Thayan Apprentice? And a wight? We've had ten of those already."), and stopping to rest for long periods of time can break the verisimilitude and reduce the dramatic tension.

You can keep the drama and enhance the verisimiltude by treating the Doomvault as an active and dynamic complex which responds visibly to the players' actions and pushes its own goals against the world around it. You can help the player characters push through the many encounters by providing them with extra resources. And you can keep the game relatively fast-pased by ensuring that the players stay focussed on the end goal and directing them towards the end.

A dynamic dungeon

The Doomvault is an active military complex. The Thayans within have a whole army and facilities to create monsters to bolster their ranks. The Thayans within are at war with the Thayan rebels keeping them inside. Do not think that the Thayans will just sit by idly as the PCs march in and start hacking and slashing their way through the Doomvault. These suggestions delay burn-out by making the Doomvault change over time.

• Alert Level and scaling encounters: A simple change to make is to the Alert Level, which is a mechanic in Dead in Thay which makes the random encounters more dangerous the longer the players spend in the Doomvault. I have found that having the Alert Level go up by 1 per hour (rather than 1 per 4 hours) gives a nice progression. For the first day or two they had basic random encounters, whereas towards the end of the adventure they were getting higher tier random encounters. The earlier encounters might be flavoured as ordinary patrols, whereas later encounters may be squads sent specifically to hunt down the party. The scaling severity of these encounters will enhance the sense of danger the players feel the longer they spend in the dungeon.

• PCs become (in)famous: The Doomvault is also occupied by intelligent NPCs. In the first day, the Thayans at large might not notice the PCs unless they make a big fuss, but by the second day word will have spread that there is a party of adventurers raiding the Doomvault. If some Thayan-aligned NPCs witnessed the PCs and lived to tell the tale, the Thayans will be well-informed about the party's composition and may start preparing countermeasures. If the PCs ensured there were no witnesses, then the Thayans will only know that someone has been slaughtering parts of their dungeon and that they need to be alert. Over time the PCs will become quite well known (and likely deeply feared) by the residents of the Doomvault. Your players will get a kick out of overhearing a conversation about their exploits or from an enemy who is cowed by their fierce reputation, and it makes them feel like their actions have impact.

• Use the dungeon's facilities until they break: Part of the goal of the PCs is to disrupt activities in the Doomvault, so keep track of which activities the PCs disrupt, and which ones they don't disrupt. For example, before the party clears out the Ooze Grottos, you can send oozes to reinforce other parts of the dungeon. Once they kill most of the people in the Ooze Grottos, though, the Doomvault no longer has the capacity to send oozes to reinforce other parts of the dungeon. There are other examples. Two of my favourite facilities were the Undying Laboratory and the Golem Laboratories. While the Undying Laboratory still stood, I had it convert some of the corpses of monsters my players killed into new undead creatures, such as skeletal trolls or zombie gorgons, which presented some unique enemies for the players to fight. As for the Golem Laboratories, having the threat of golems looming over the players' heads was a great motivator for them, even if they fought only a few flesh golems. Note that if the players clear out an area, the dungeon loses the ability to send this sort of enemy. If my players had braved the Golem Laboratories and prevailed, they would have been safe from golems from that point onwards.

• The dungeon regroups while the PCs rest: This is explicitly stated in the adventure module, although you might initially be at a loss as to how to make the dungeon recover without sending the PCs back to square one. Whenever the PCs take a long rest, you can replace some of the non-unique creatures in a room. Unique and high-level creatures, however, should stay dead. Consider the functioning of that room and what staff are needed to provide the minimum functionality. Also consider what the Thayans have to replace lost manpower. They have a sizeable number of humanoids, but that number is still finite. However, they also have several necromancers, so any slain humanoids can be raised as skeletons, zombies, wights, deathlock wights or dread warriors, providing ongoing supplies of those creatures. Additionally, the Doomvault can create oozes and golems and summon demons and aberrations, so these are also renewable sources of minions. Furthermore, the Thayans may reinforce strategic locations once they learn of the party's actions (golems are a favourite of mine), encouraging the players to push through and clear out an entire sector without taking too many rests.

• Make resting unsafe, and the dungeon has its own goals: In the case of Dead in Thay, it is stated that the goal of the Thayans is to survive and to kill the party. However, I think this overlooks another goal, and that is to break the siege which the rebels have placed on the Doomvault. It would probably take a day tops for the Thayans to realise that Syranna has captured the gatehouse and is trapping them inside their dungeon. With the party slowly overwhelming their defences, the Thayans will want to break through to the gatehouse so they can deny the party their resting spot and get reinforcements from outside. After many sessions, I had the party walk into the gatehouse in time to witness the conclusion of a large battle between the rebels and the Doomvault Thayans. The rebels had won for now, but this event signalled clearly to the players that the gatehouse was not as safe as they once thought and that they needed to hurry up and finish the adventure. I planned for a series of progressively harder encounters set in the gatehouse for each time they long rested after that, although my party managed to finish the adventure before that. They pushed for a rapid conclusion because they felt that long resting was dangerous. This sort of tension adds greatly to the feeling of a "fast-paced assault". However, I caution that this tactic should not be used too close to the start of the adventure, but rather be used to hurry up the party when you get closer to the end.

Assist the party

The Doomvault is taxing on the party's resources, especially for small parties like your own. As you have discovered, you quickly blow through a whole adventuring day's worth of encounters in just a couple of in-game hours. The above points help you motivate the party to keep going despite the strain on their resources, but there will still come a point when the party simple doesn't have any resources left to keep fighting. As such, you need to help the party out.

• Give them care packages: In Dead in Thay, the party is asked by the rebellion leader Syranna to raid the Doomvault. Syranna has a vested interest in the party's rapid success, so Syranna would leverage whatever resources she can to aid the party. To provide healing, I gave the party a liberal supply of healing potions. To deal with casters who were running out of spell slots, I gave them spell scrolls. If you want the party to go for longer without long resting, give them consumable resources which effectively grant extra uses of their limited use abilities or otherwise supplement them (e.g. spells, healing). Then I had Syranna tell the party not to come back until they had finished using those items.

• NPC Allies: A simple remedy to a small party is to make the party larger. The Doomvault contains numerous NPCs who have the potential to become allies. Some of them might require a long rest to get back to full health, and some might need to be assigned a stat block, but they can be used to aid the party if they are under strength. You could also bring in NPC allies from previous adventures. Don't give them too many NPCs at once though, as that would slow down combat and take the spotlight away from the players. You can let the players keep some agency and take the pressure off you by letting the players determine the NPC allies' actions in combat (although you keep veto power to stop them from acting out of character). For Dead in Thay in particular, if the party lacks proficiency in Arcana, an NPC ally goes from useful to essential because many things in this adventure require Arcana checks.

• Recruit more players: Ideally you want a large party for this adventure, not a small one. If at all possible, if you have a small party you can attempt to increase the size of your group by getting more players. This is not always possible (I wasn't able to do this), but if it is possible it is preferable to NPC party members.

• Facilities for restoration: There are a couple of places in the Doomvault which provide benefits to characters which would otherwise require a rest. There is a magic pillar in one place which is an unlimited source of healing (with a small risk associated with it). There is also a pool in one place which grants the benefits of a short rest once per ten days, although that was woefully underpowered so I house-ruled it to be the benefits of a long rest (ignoring normal time limits) once per day. Such facilities allow the party to adventure for longer each day. You can guide the players to discover these facilities, and remind them of their existence whenever you think the adventure would benefit from their use.

• Let them shop: There isn't much loot in the Doomvault, but there is some. In my campaign I had established the existence of a magic item merchant as a recurring NPC, so I had her show up during the party's extended long rests for them to trade if they wanted. This let them collect new assets which would help them in their adventures, and let them liquidate loot they had acquired from the Doomvault without having to wait for the end of this very long adventure. You will need to figure out something which makes sense for your campaign. In any case, your party will likely have many hours of downtime each day as they recover from adventuring, which can be spent shopping outside the Doomvault or crafting.

Goal-oriented raiding

In the massive Doomvault, it is very easy for the players to lose sight of why they are raiding this dungeon and just get stuck in the grind of mega-dungeon combat. But the party does have a goal, and it is your job to make sure the players stay on track.

• Plentiful clues: One of the party's objectives is to determine the location of the Phylactery Vault. For this, you need to give out lore. The adventure module has a list of items of lore which you are meant to give in order and certain NPCs who can give a set number of pieces of lore. However, I recommend not sticking too strictly to this system. The items of lore are good, but some NPCs would know more than others, so you don't need to stick too closely to the order or the particular number of pieces of lore. The more clues you give out, the faster the players will be able to finish the adventure. I would lean towards being generous with sharing of lore.

• Tell them to stay on track: In Dead in Thay, this can be done using the NPC Syranna, who the party answers to. Have Syranna encourage the PCs to keep going and not rest every 15 minutes. Have Syranna remind the party of what they need to look for. Have Syranna point out major leads the party should follow. Have Syranna ask the party regularly for their progress. Have Syranna intercept communications or track movements within the Doomvault to give the PCs hooks to go to certain parts of the Doomvault. Constant prodding and reminding will help keep the goal at the forefront of the players' minds.

• Guide the party to more interesting areas: Unlike many dungeons, which have a fairly linear design, the Doomvault is almost a sandbox with 7 different entry points. However, some sectors are more interesting and plot-relevant than others. My party started by going into the Ooze Grottos then progressed to the Forests of Slaughter, which are two of the sectors with the fewest NPCs which can be interacted with outside of combat and next to no hooks to the greater goals. The players rather quickly got bored of these sectors and the game became a grind until they moved on. I would recommend providing the players with recommendations for good places to start or go which lead them away from the boring areas and into the interesting areas which have more plot hooks.

If you apply these points of advice, they should help keep the adventure from getting stale and push the players to progress through the adventure more quickly. While some of this advice is specific to Dead in Thay, it should be applicable to similar adventures too.

For reference, I'll give you a few of the details of my time running this adventure. For most of the adventure I had three players at each session, with a wizard, a paladin and either a druid or a barbarian. It took us 14 four-hour sessions to finish Dead in Thay, although your mileage may vary. The party took a total of 4 long rests throughout the adventure (although without the magic long-rest-granting pool, they would have taken 5 long rests). The party started at level 9, and by the end, before the final boss, the regular players had reached level 13 (although if I had a larger party they would have probably only reached level 12).