it was an iPhone 13; I don't know if the battery is "really" connected or it's just how they do the repair; he just mentioned that the back cover and battery would be part of the logic board replacement.

(I still am not quite certain why they didn't just give me a refurbished phone and later repair mine as a refurb, but whatever)

No, and in the newest iPhone, 14, it was made easier to replace the battery:

https://www.ifixit.com/News/64865/iphone-14-teardown

I think X to 13 models had particularly difficult battery replacements.