I agree with you that arranging play dates in grade 7 isn't the way. Your son has to be responsible for his own relationships. He might need some help with awkwardness or social anxiety, and you might talk to your pediatrician about finding some professional resources. I agree with you that it's okay to be "who he is" but only if "who he is" is a complete and happy person. If your son is not functioning in required situations, that's a problem. What strategies should be employed, or even what "diagnosis" is at play here (if any), need evaluation.
In general, I think it's not acceptable for kids just to be content to be at home with computer games and only a younger sibling. Being a homebody is okay, within reason. But it's not enough. They have to learn to be a functioning member of society on some level. The best thing would be if your son were required to join some sort of after school activity where he would broaden his social skills with kids with similar interests, as well as begin to develop some interests outside of academics. The isolation of computer games cannot be his only activity. So I think it may be time to say that he's growing up and he has to push the envelope a bit, and move beyond his comfort zone. I'm not into over-scheduled kids at all, believe me - but they should have to do SOMETHING that interests them or at least which benefits others in some way (the school community, the town, a religious organization, something).
However, if you are unsure, I think you should contact a professional to find the best way to do this. Your son might benefit from joining a club. He doesn't have to play a sport or get "out there" on stage for the school musical, but being a part of the chess club or the social action committee or the stage crew for that musical might be better. But whether forcing him to do it and just continuing the mom-to-son talking (which doesn't seem to be working) should happen first, or the professional help should happen first, is a tough call and not one that those of us outside the situation are prepared to make. But I wouldn't let it just go on forever since you're quite clear that you can see it's not working. If he's "just an introvert" and will grow into a contented person in time, that's one thing. But if he's headed down a path of unhappy isolation and not being able to function in a college or a job, whatever he will do a age 18, that's not okay.
ETA: Just checked your prior posts - this is the same kid who had a tantrum over blood work, right? So there's something going on with him, I'd say, related to fears and anxiety.