I don’t think any of the things you listed here as benefits of counting encounters can’t also be said of XP. Or, rather, I don’t think these are benefits of either reward system, they’re benefits of deciding what reward to give after the encounter is over instead of deciding in advance when designing it. There’s no reason you couldn’t do that with XP, and using XP only provides more granularity of how much to award after the fact.
Only if lethal combat is what you award XP for.
[image snipped]
CR is a notoriously loosey-goosey system. The dirty little secret is, combat XP is just as arbitrary as non-combat XP. Sure, there’s a bit of math that allows you to more reliably eyeball how likely a combat encounter is to kill the PCs, but eyeballing is really all it is. And most DMs I hear from ignore CR anyway. If you trust yourself as a DM to eyeball combat encounter difficulty, why not trust yourself enough to eyeball non-combat encounter difficulty?
That feeling of “realness” is purely illusory. You’re right that most DMs undervalue non-combat challenges, but I think trying not to do that is a much more direct solution to that issue than not using XP.