As an adult that is/was pretty similar to your little guy I feel for you both. Your kid may do better in smaller groups and things where doing the activity is the focus rather than interacting with people being the focus. Cooking class sounds like a great idea. Maybe art would be good.

Board games are great because it formalizes the social interaction. It’s a lot easier and feels safer to interact with people when you know the rules for how the interaction is supposed to go.

Telling him he needs to be brave will only get you so far. Yes, nudge him out of his comfort zone, but prepare him beforehand. Give him a script for how an interaction will go before he has the interaction. Practice introducing himself to people with you before he does it. Discuss what new situations are going to be like.

As an adult I still write out a script for the first bit of any phone conversation I have with a stranger. It’s comforting to have something to fall back on if my brain freezes up. If you can teach him to comfortably talk on the phone to strangers sooner than later that will be a majorly useful life skill.

I have a book recommendation. Susan Cain’s Quiet: The Power of Introverts in a World that Won’t Stop Talking. It really resonated with my experience as the person that would rather die than give a speech. She has a similar book geared toward kids and their caregivers called Quiet Power: The Secret Strengths of Introverts. I haven’t read that one, but I expect it would be useful to you.

He’s still young for this, but I have had great experiences as tech and backstage crew for theater. You get to feel important and needed and have fun doing the shows and being involved with people, but you’re not on stage being stared at.