The "regret not getting to know each other" is definitely something that I've heard from recent YC alumni.

As for weekly gatherings--I think that might be a bit distracting for the teams there, but I can see occasional gatherings being valuable.

This. Thinking and problem solving is a social process, the human brain is designed to be fueled via lots of nuggets of social. Theres a huge amount of variability in what sorts of socialization different folks need and what form it needs to take, but it's absence creates a terrifying inability to think clearly and execute.

Foam earplugs plus around the ear headphones solve most hearing other people distractions. :)

Point being, if my group is in YC this summer, we're definitely looking for a social living space

It could be distracting but a weekly (small) gathering would be pretty beneficial. Teams could get feedback and their pitch down. It'd be like "office hours" with peers and a couple of beers. I am sure you will experiment to see what works.