Yes, this build is competitive. Compared with a strength based dueling paladin, you'll actually be better off in many ways. Though you can't compete with a great weapon fighting paladin in damage. To illustrate this: let's compare some general build options.
Let's assume there are two paladins: Strong and Dexter. Strong and Dexter have the same stats except that Dexter has dexterity equal to Strong's strength, and strength equal to Strong's dexterity. This means that for the purposes of attack rolls and damage modifiers, the two are exactly the same. For that reason, I'm omitting modifiers when I make damage assessments below. I'm also not including magic items because with enough magic, you can make anything work.
Sword and Board
Dexterity is very competitive here, if not the best choice. In this case, both paladins take the dueling style and wield a weapon in one hand and a shield in the other. The highest damage one handed weapons all deal 1d8 damage. For Dexter, that means using a rapier. For Strong, it could be a longsword, warhammer, battleaxe, or rapier. Regardless, because the damage die is the same, and the paladin's relevant ability score modifiers are the same, their damage output is identical. Eventually, when Strong gets plate armor, they will have 1 higher AC than Dexter does (18+2 vs 17+2, as you have noted) Strong also has options to deal all three types of physical damage (slashing, bludgeoning, and piercing). Dexter will only be able to deal piercing damage, but their initiative, stealth, and dexterity saves will all be higher than Strong's. Dexter also has much better ranged attack options since they're as effective with a bow as with their rapier.
In return for being 5% easier to hit compared to Strong with plate armor, Dexter will be better at range, go earlier in the initiative more often, and make their dexterity saving throws more often than Strong does. Honestly, that's a pretty fair trade off.
Great Weapon Fighting
Dexter can't compete with Strong's damage here. Dexter keeps their shield and rapier. But Strong takes great weapon fighting and a heavy weapon. Their damage die goes up to either 1d12 or 2d6, depending on which weapon they choose. And they get to reroll 1s and 2s on that damage. Using the results of the How much damage does Great Weapon Fighting add on average question, that works out to roughly an extra +1 damage on average. So, strong will be dealing about 8 damage per hit before modifiers regardless of which weapon they choose. Strong could also take the Great Weapon Master (GWM) feat to further increase their damage output. Dexter is still dealing about 4.5 per hit before modifiers with their rapier and doesn't really have any feats that will consistently increase their damage to Strong's level. So, Strong is dealing twice as much damage as Dexter on average. However, Strong loses their shield. So even with plate armor, Dexter now has a 1 AC advantage (17+2 vs 18), making them 5% harder to hit than Strong. And Dexter is still better at initiative, ranged attacks, and dexterity saving throws.
Two-Weapon Fighting
Here's an interesting option. If Dexter has 13 strength, they can take one level of fighter to pick up the two-weapon-fighting style. They also take the Dual wielder feat so they can use two rapiers. Now, Dexter and Strong have the same AC (17+1 vs 18) and Dexter deals an extra 4.5 damage on average with their off hand attack. up until 4th level, Strong will deal about 8 damage per hit and Dexter will deal about 9 damage over two hits. So Dexter hits harder than Strong! As long as both attacks hit anyway.
The issue is that once the paladins hit 5th level they get an extra attack. Strong now deals 16 damage over two hits (8 damage twice), and Dexter deals 13.5 damage over three hits (4.5 damage thrice). And that doesn't factor in the bonus damage that GWM offers Strong. At this point, Strong hits harder than Dexter does, and again Dexter can't do much to catch up.
Final Thoughts
This is a very quick analysis. Enough to get the point across I think. There is no way for a Paladin using dexterity to deal the same damage as a paladin that's focusing on dealing as much weapon damage as possible. If you want to deal maximum damage, you have to go with a strength paladin. But dexterity gives you much better survivability. Dexterity saving throws are very common, and can deal some really high damage (think fireballs, lightning bolts, and fiery dragon breath). And with higher dexterity you can go earlier in the initiative and position yourself where you need to be as soon as possible. Plus the large damage bursts that paladins are known for because of their smite is, as you have pointed out, not dependent on weapon damage. It's just based on spell slot level. All of that is to say, the survivability and flexibility is something that will definitely serve you well and I personally think is a reasonable trade-off.
The answer to both of your questions (this and the parallel one) is the same:
Stop giving all of the NPC attackers the same initiative.
That step right there eliminates the focus fire problem. That choice is the first element of a two part solution. You are the DM and you have this choice.
The Cleric is getting to 0 HP quite frequently, though, due to burst
damage caused by all the NPCs attacking at the same time (as stated in
the other question) or simply being unlucky.
If you break up the attackers in to smaller sub groups, this burstiness is mitigated.
Now for the experience based part of the answer.
The DM in our first campaign in 5e taught me a valuable lesson; he broke the NPC enemies into groups once the number of enemies were larger than 3 or 4. He also had leaders, or different kinds of monsters, roll a separate initiative. All but one of my other DM's have done the same. You can watch the difference it makes at the table. A side effect of this is that the battle gets to look a little more like a tennis match: a couple of their guys attack, a couple of our guys attack, etc.
Example: seven goblins and two ogres fighting a party.
- Two groups of goblin initiative (a clump of 3 and a clump of 4) and
either separate init for each ogre, or both the same.
For a level 2 party, I'd spread it out because Low Level D&D 5e is
swingy.
My advice on "the dice are fickle" applies to this also. This approach spreads initiative out so that party actions are interspersed with enemy actions. It's still swingy, but it is less bursty for one side also.
Second element of a solution: a primer on tactics for your players
Sit down with your players and discuss tactics. While each group has its own opinion on how close to "combat as war" they want to get, your choice of playing "smart monsters" (IMO a good one) means that the players have to learn
- how to apply their tactics As A Team, not as individuals
how to play smarter not harder.
Sometimes, it takes some DM coaching to get them working in that direction.
A tactical point to discuss with your players for this particular problem at 2d level: what are the spell casters doing to slow down the enemy? What crowd control or "dividing the enemy" measures are they taking?
You are the DM, so coach them. Almost every DM I've had over 4+ decades with this game has done a little, or even a lot, of coaching. It comes with the role.
As a final observation on your problem: if, even when faced with the facts, your player still thinks you are out to get him, there is only so much you can do as the DM. Some people are simply like that. My suggestions above may help mitigate that so that this player's perception changes.
Best Answer
Be A Diva
You are essentially looking for some way to handicap your pure-kill potential, and you're looking for some way to act that will generate that. There are an unlimited number of ways to do this, here are some that are famous and/or have worked for me: