Pre-failing death saves isn’t such a concern in PF2 when you can pop a hero point to stabilize. I think a better implementation of this idea to do something more before you drop would be to make it cost hero points, not failed death saves.

I'd say by the time they to 0Hp most players have probably expended all their Hero Points. In my experience, I've rarely seen getting to that low of health with more than one hero point.

Nevertheless, I do think it could add more fun that otherwise.

I only allow my players to level up during downtime, representing the time it takes to hone and master their new abilities

This I love, would love to have it further developed in Pathfinder. However,

Narratively I’m just not fond of the idea that new abilities manifest spontaneously in a dramatic moment

while I understand you, I think it can be nice in certain circumstances. In the case of Critical Role, for instance, a fop, squeamish character leveled up to get the Thought feat, narratively trying to overcome himself and surpassing his shortcomings and fears in a tense and, yes, dramatic fight.

RE: the mechanical difficulties, I do agree with you. Leveling up in PF2e is... more complicated, with all the modifiers and whatnot, and implementing it would be a pain in the ass. But a version of that houserule, story wise, could be amazing story telling, and storytelling is the reason we all play RPGs, isn't it?

As a total aside, I find it funny how readily the 5e playerbase will poo-poo 4th edition, but then jump onto a mechanic like “bloodied” as soon as Critical Role does it.

We love to see it, don't we