The Crimson Binome
Hero
While it is true that we don't need rules for what happens off-screen, we do have rules for what happen on-screen. If the off-screen activity varies noticeably from the on-screen activity, then that's an inconsistency within the world.Yes, in the background, the character recovers from his injuries in an appropriate amount of time. That doesn’t need to be governed by game rules though, because it isn’t part of the game. It is part of the narrative, and can (and should) be governed narratively.
Recall that the rules of 5E explicitly condemn meta-gaming. You aren't allowed to treat an NPC differently than a PC, just because they're an NPC.