I'm a big fan of the use of a dying condition, or at least being reduced to 0 hitpoints being referred to as dying, just because it's so clunky to accurately refer to it at the moment.

The thing I can see in yours that is perhaps an oversight (perhaps planned) is that anything that is designed to modify all saving throws such as the monks proficiency in all saves or the paladin's aura of protection would make succeeding by 10 or more easier in a way that's not currently covered by requiring a natural 20. (Both of these abilities currently apply to death saving throws but do not make natural 20s easier of course). Also bless, and any bonus to all saves from magic items work on death saving throws too. This also impacts the ability to fail by more than 10, making it effectively impossible without a different homebrew feature creating a penalty to the roll.

Seperatly you call it a check, which means it wouldn't be impacted by these saving throw alterations but would be impacted by anything which alters all ability checks such as the bard's Jack of all Trades, or more concerningly, the combination of the rogue's reliable talent and any way to gain proficiency in it, which is basically an automatic success.

There are a few currently niche cases where characters gain a bonus to all saving throws or specifically death saving throws which is intentionally factored into the power of the feature, and makes them exciting and useful, that are either hugely buffed (depending on how you rule their use) or totally discarded in replacement to other features that weren't balanced around this ability.

Something I do love in this is the ability to introduce the wounded condition outside of being unconscious. Imagine how scary something like a nightwalker would be if it's aura didn't do necrotic damage but instead just forced a con save Vs getting a point to your wounded condition.

Personally the way I'd handle this is to make dying a condition that is basically identical to the death saving throws mechanic currently in 5e, but have it reset when you gain hitpoints by any means, if then disconnect being unconscious from it entirely at a mechanical level and just say if you gain hitpoints when unconscious you may choose to instantly end the condition. This would mean everything that currently works in the game to offer a bonus to death saves remains, and in very rare cases, you may make death saving throws while not unconscious, either counting from there when you fall unconscious or dying while on your feet at 3 failures.

I'd also change taking damage while unconcious to just force you to make a threatened dying save, which is just like a normal death saving throw except you don't mark a success if you get 10 or more, you may only fail. This means that you can wail on an unconscious PC without worrying about killing them without agency. I'd probably also make the spare medicine checks function as a protected dying save, where you can't fail and can only gain a success on 10 or higher.

Edit: I didn't know this was pathfinder, I just assumed it was your homebrew for 5e.