I totally agree that's really important to keep a balance and nurture irl relationships. But, one has nothing to do with the other. It's supposed to be at working hours in which you would still be at the computer anyway. And from my perspective after working remotely for a while is that there's a point where you want to know more about your peers and share some non-work time.
Someone over twitter shared "Keep Talking and Nobody Explodes " as a great game to play among peers. Still, I totally agree that trying different games is a great idea.
The idea is that you can do it in working hours and it would be totally optional. Probably a good idea is that you have some hours to spend on social interaction, and you spend them however you want it can be playing the game with coworkers or even meet with people outside the org. Anyway, in most remote roles you manage your time.
Like someone pointed out, seems like a problem you would have either way with that kind of person (bad losers). Still, competition can get though, that's why one of the ideas is to compete against other teams/companies' teams.