For stat raising buildings like hot springs, do you need officers garrisoned there? Sounds like it from the player guide linked but I can't find info anywhere else
I can't remember, but I've found the effects to be pretty much negligible. Full disclosure, I play very, very aggressively.
The problem with building upgrades is that, other than the ones for money and soldier generation, it's generally not worth putting officers on them to guard/benefit from them. You probably need to station some soldiers at the border at any given time, which will move as you expand, and soldiers taking a break from the frontlines will spend only a few turns at most to recover. Higher stats will see gains more slowly and lower stats aren't going to help you win any major fights. It's much easier and more efficient to simply capture and convince strong officers to join your army and swarm unguarded/less guarded cities as often as you can (within diplomatic reason). Your stronger officers can be used to distract and funnel the enemy's stronger officers to one location, and the AI will never willingly have them sacrifice half their officers to retreat, allowing your weaker officers to freely assimilate the other cities in their control.