I generally have a hard time connecting with people like OP but found that I was able to find good climbing partners outside as opposed to in the gym.
I now do crossfit and while I know it's not for everyone, it's a decent community. I still don't talk to folks in the gym, I don't want to but I like that we're all in it together and pushing ourselves pretty hard. I feel connected in that way.
I would really not like a stranger tapping me on the shoulder in the gym. That's my "alone time". That's just me though.