Why would that be? Death's door doesn't mean you've been physically injured. What does death's door mean to you?
Well, in most D&D situations one usually gets to death's door through being repeatedly hit by some sort of weapon (or weaponized body part), or by being repeatedly shot with some sort of missile(s), or by being burnt/frozen/necrotized/shocked/etc. by some sort of spell effect(s).
None of those are going to fatigue you to death. You can dodge...until you can't, and one of 'em finishes you off, causing actual physical damage of some sort in the process.
I guess in a non-meat hit point world it means the edge of exhaustion or similar. Why does that necessarily mean a night's rest can't cure that?
Can't speak to your experiences, but I've never known anyone in the real world to be on the brink of death one day and feel right as rain the next. I'd like at least a modicum of that reality to bleed over into the game.
But I get what you're saying: it would be interesting if there was a penalty for getting to the point of having to make death saves. Maybe that imposes a penalty on the max hit points you can heal on a rest?
Now you're talking!
The way I do it (in an older-school system) is that if you've been near death recently nothing can cure you above a pretty low total until some time has passed, with the length of that time being determined largely by how far down you went. Said time can be shortened somewhat if all you do is rest. There's more to it (it's all on a body-fatigue point chassis) but that's the very basic version.
For your idea, maybe for each death save you have to roll in a day (whether it's successful or not) you can't recover 10%* of your hit points that night. So, if you roll 4 death saves in a day, the next long rest will only get you back to 60% of your total; with the remainder only recoverable either through curing (at half effect) or through another long rest that cannot be in the same calendar day. It maxes out at the %-age, thus even fi you somehow have to roll 14 death saves in a day you'll always get back to 10% of your total tomorrow.
* - or whatever % value seems to suit.
Maybe it applies a level of exhaustion?
Could do, but I think I like the limited-recovery idea better.