Rules as intended, it's the DM's choice. Jeremy Crawford, lead rules designer of D&D 5e, was asked this exact question, and said that it's up to the DM. Not what you wanted to hear, but you did say rules as intended was the best type of answer. Weirdly, he has since said that the steed leaves gear behind.

RAW, on the other hand, would suggest that your steed can't take anything with it.

When the steed drops to 0 hit points, it disappears, leaving behind no physical form. You can dismiss your steed at any time as an action, causing it to disappear. In either case, casting this spell again summons the same steed, restored to its hit point maximum.

"It" disappears, not "it and everything it's carrying". And when you cast the spell again, "the same steed" appears, not "the same steed and everything it was carrying last time it disappeared".

It's also worth considering that "you summon a spirit", which "assumes the form" of a steed. It's not a physical steed that you're summoning and dismissing, so it doesn't make all that much sense for you to be able to summon and dismiss physical objects along with it.

Yes, the summoned mount follows the normal rules for mounts

Per the description of find greater steed:

You summon a spirit that assumes the form of a loyal, majestic mount.

As you said, mounts can be handled in one of two ways: you can control the mount or you can allow it to act independently (per the rules for mounted combat):

While you’re mounted, you have two options. You can either control the mount or allow it to act independently. Intelligent creatures, such as dragons, act independently.

The summoned steed is a mount and thus follows this rule unless something else says otherwise. The spell actually says:

You control the mount in combat.

Which means that you have full control over the animal if you wish. Jeremy Crawford says that you can even choose to let it act independently if you wanted:

Find steed / find greater steed—when you ride the mount in combat, you decide whether it follows the rules for a controlled or an independent mount.

There are no other passages from the spell dealing with control of the steed and certainly nothing that would imply a special method of control that was unique to your summoned steed. Remember, spells do only what they say they do. There are no secret rules. If there was some special method for controlling the steed, the spell would say so and detail how it worked.

So, the spell grants no special control method for summoned steeds.

There is also nothing in the description of find greater steed that has any friction or conflicts with the normal rules for mounted combat.

Only find steed says anything about fighting as a seamless unit

Contrary to your question, only find steed has the following passage:

Your steed serves you as a mount, both in combat and out, and you have an instinctive bond with it that allows you to fight as a seamless unit.

The only thing like it that find greater steed says is the quote about controlling the mount which I quoted above.

However, even if that text was in find greater steed, fighting as a seamless unit is exactly what the controlled mount option is. Controlled mounts normally have to be special creatures that were trained to accept a rider.

Jeremy Crawford has clarified and made it clear what that passage was intended to mean:

The spell says that you and the steed fight as a cohesive unit and you can communicate with it and it serves you.

Really what that means is, it is up to you [...] whether to control it or let it act independently

Your bond with your steed allows you to fight as a single unit with it (controlled), but "allows" does not mean that you always have to be fighting as a single unit. Hence you can still have the mount act independently and be two individual fighters as opposed to (essentially) one.

Regardless, the conclusions for find steed are the same as for find greater steed: they follow the normal rules for mounts.

The mount only acts "randomly" if you let it act independently

You control the mount when you want to. When you control it, it doesn't act randomly. If you don't want the mount to act in a way that you as a player don't control, then you should choose to always control it. However, it is worth noting that even when the DM is controlling the creature, it is not going to be acting randomly. It is still going to act according to it's intelligence and what its and possible your best interest.

Having a summoned steed has plenty of benefits over a normal horse

The spell itself grants many advantages to using the spell to summon a steed as opposed to buying a horse:

• The steed is also affected by spells that the rider cast (while mounted) that target only the rider
• Telepathic communication with the steed
• Not having to purchase a steed
• Being able to resummon the steed if it dies as opposed to having to replace it
• Being able to control the mount without training
• Increased intelligence
• Option to choose the form of your steed

The first bullet alone is huge for spellcasting and allows for some really neat synergy in battle. But as a whole the summoned steed is unambiguously better than mounts you can find or buy.

Form matters a lot

Not only do the mounts here have better stats than a common horse, some of them have flight which is a huge benefit in many cases. Having at-will flight is so good it is often seen as being over-powered and ways to get it are very limited.

Not only that, but the cosmetics matter. Being able to choose the form of your steed is an advantage in and of itself. Where else are you going to find a tame saber-toothed tiger to ride? Or a rhinoceros?