As PC’s progress, falling to 0HP in combat gets less and less meaningful. So I have used a rule that whenever a PC is at 0HP at the end of their turn, OR fail a death save, they take a level of exhaustion. It makes the 0HP yo-yo more dangerous, and makes it so “death” has some longer term consequences.

I’ve got an Oath of the Ancients paladin in my group. I play it as if the thing behaves like a spell, we treat it as a spell for the purposes of this feature.