Manbearcat
Legend
The fighter's friend is hurt, even dying. The fighter prays to the gods to heal their friend. How is this action resolved? If it's a Religion or similar sort of check, what's the DC?
Well...I think you have your answer to "does tactical infinity actually exist in D&D for most tables?!"
And I don't agree that you need a giant reservoir of fiction to conceptually resolve this. I've dealt with stuff like this in so many systems with so much divergent fiction and game engine dynamics that I'm just going to sub in some ideas for what this might look like, consider mechanical implications, and wing an answer.
Alright, as you know, I only run 5e very sparingly for a group that isn't a group that I typically run games for. So take this "not full time 5e GM" answer with in mind.
What I would likely do is the following:
* Medicine Kit makes stabilizing character's trivial, no check, so that is a problem right there. Getting people back in the fray is pretty common too so that is also a problem. This isn't that big of a deal.
* Wis (Medicine) also lets you stabilize without Medicine Kit. I would typically call this DC 15.
* If this was in combat and was to either (a) get an ally back into the action or (b) stave off a final Death Save, it would be an Action. Otherwise, its just freeform. If the use is for (a) I would make the DC 20 and the PC is up and goes back to 1 HP or if its (b) I would make it DC 15. Regardless, it would be Cha or Wis (player's choice depending on how they feel about the nature of their appeal or their relationship to the Gods...is this citing liturgy and resolving ritual with precision or is the strength of belief and compulsion to get the god to act, dammit!) w/ Religion.
* I would use the Basic PDF Fail Forward handling for this. We would stake a HD of your HP which you can't get back until you pray again (giving thanks) during a Long Rest. Further, create a Bond w/ this PC/God/Faith on a success or some kind of Flaw/(negative) Trait on a failure.
So, it looks like this:
Success = (a) or (b) above occurs + take Bond w/ this PC/God/Faith.
Failure = (a) or (b) above occurs + lose 1 (Class) HD worth of HP which are unrecoverable until Long Rest prayer + take Flaw/(negative) Trait signifying the trauma/hardship of the event.
So...yeah.
The big problems for 5e here are (a) "this just ain't that big of a deal mechanically" (combats aren't difficult and getting people stabilized or back in fights isn't difficult) and (b) the machinery of BIFTs and toothy interaction with play isn't robust (but they do a smattering of non-ribbon work).