I've thought about it a lot. In the end I think your therapist got it right. It's all about circumstance. You have to live in a place that promotes daily interaction with people. Dorms and fraternities promote that sort of thing not just by being in the right location but the physical geometry of dorms and frat houses encourages people to meet and hangout. The rooms in dorms and frat houses are so small that people just end up using the larger shared lounges for dinner, studying, and chilling in general. It forces and promotes daily interaction with large groups that you'll never experience in any other setting.
I think the reason why the sports thing or the young professional stuff didn't work out was because it needs to be a daily thing. Connections will always feel awkward especially for introverts because it's just not happening often enough.
Anyway back to my point of the living space. Outside of colleges I've seen this type of living space in just two places. The first place is called Cooperatives. People in coops live in the same building, do shared chores and have communal dinner. Like frat houses these places tend to be pretty ghetto but it's never lonely. The other place is a retirement home. If you've never been to one, retirement homes actually have shared spaces and dining halls where you can literally observe older people having the same kind of social interaction and politics you experienced in college living spaces. The irony of it all is the same social experience young people get living in dorms near the beginning of their lives, they'll get to experience the same thing when they near the end.
So I guess the solution is to either live in a coop, wait until your old enough to be placed in a retirement home, or get married. Unless anyone has any other suggestions?