After reflection hot seat might have just been a gap/brain fart for me. It didn't impede my experience. It feels weird to clarify but I think it was simply useless feedback by me in the first place.
But it's a good point though. The term is indeed archaic and not that widely known anymore as games nowadays don't really have such mode. Hell, even all local multiplayer modes are a bit of a dying breed unfortunately. And if someone doesn't know the term, there is a risk of them clicking on it to see what it does, if it toggles something etc. Exactly what I wouldn't want and that's why the button is smaller than the play button and so on. So I think more clarity would have been good there. If the button said "hot seat multiplayer", at least most people would get the point that it's some sort of multiplayer mode.