Identify

1st-level divination (ritual)

You choose one object that you must touch throughout the casting of the spell...

If you instead touch a creature throughout the casting, you learn what spells, if any, are currently affecting it.

Although it does not mention curses specifically, looking at a cursed item, say Armor of Vulnerability:

Curse. This armor is cursed, a fact that is revealed only when an identify spell is cast on the armor or you attune to it.

Since it was brought up as a comment; the duration listed in the spell is how long the caster has to implant the memory.

On a failed save, the target becomes charmed by you for the duration.

While this charm lasts, you can affect the target’s memory ...

If the spell ends before you have finished describing the modified memories, the creature’s memory isn’t altered. Otherwise, the modified memories take hold when the spell ends

Once done, the receiver is affected by the spell until it is dispelled. Fireball has an instantaneous duration, but the long lasting affect of burnt flesh and lost hit points.

Your RAW interpretation is correct

The interaction with lesser restoration

When casting lesser restoration on a target affected by feign death, the RAW result would be the same as casting it on a target affected by the likes of blindness/deafness, contagion, sunbeam, or any other spell imposing the blinded condition: the blindness is ended.

This is odd, though, because with feign death the target is in a state of blind and immobile catalepsy verging on death because they opted into it, not because they were harmed. My guess is that the designers didn't consider that anybody would bother to cast lesser restoration on the same target (or didn't care). The RAW leads to some narrative dissonance that I suspect isn't intended, which I'll address in a later section.

The interaction with freedom of movement

When casting freedom of movement on a target affected by feign death, there's no condition being ended. Rather, freedom of movement effectively suspends one of the effects of feign death.

Here's the relevant text from freedom of movement:

For the duration, ...spells and other magical effects [can't] reduce the target's speed...

And from feign death:

For the spell’s duration... [the target's] speed drops to 0.

In this case, we need to consult the rule for overlapping spell effects to figure out how they should interact (see Combining Magical Effects from the Spellcasting section of the Basic Rules):

The effects of different spells add together while the durations of those spells overlap.

So the RAW result is that the effects add together. But what does it mean to "add together" two directly contradictory effects? Since they're mutually exclusive, one spell has to supersede the other (or they have to cancel out, thereby imposing no movement adjustments on the target, which in this case is equivalent to freedom of movement's benefit anyway). So which one should the DM choose? Well, the entire point of freedom of movement is exactly to counteract effects like this!

Once again, by RAW, the result is that the effect of feign death should be superseded. Yet again the RAW leads to some narrative dissonance that I suspect isn't intended, so let's address that now.

How to resolve the dissonance

I have no evidence that the RAW result isn't intended. However, you're asking the question because, like me, it just feels like it shouldn't work this way. As you said in your question, "this doesn't quite seem like it's the intent of the spell." So let's operate under that assumption: that there's some dissonance to work through.

At my table, feign death would supersede the other spells, because my gut tells me that when you opt into its effects you don't get to have your cake and eat it too. At your table, you're free to use the RAW if you think it promotes more interesting gameplay or narrative (magical espionage action: "huh? what's that? it's just a body"), or you're free to share my gut feeling and declare that feign death supersedes. In any case, it's always up to the DM to run the game in a way that makes sense for them and their table, so you should do that.

My suggestion is to err on the side of the ruling that seems most consistent in your game's world and for the tone of the story you're trying to create. What works better for you: that a cataleptic creature indistinguishable from dead can see and move or that it can't? Your answer to that question will solve your problem.