My thoughts:
Beavers (runs about, doesn't engage or just hides): Not surprised. So would I. Massive social interaction - eye contact, lots of people speaking (we can't pick out just one voice), no clue who's who, no clue what to say or do unless it's really clear and structured. I'd want to hide in the cupboard and not come out.
The swimming club:
Disgraceful behaviour by them. But me and swimming is really, really difficult. The water is SO cold and it 'hurts', the echoes are like gunfire, the swimming costume digs in, people bump into me which is painful, and I'm not co-ordinated enough to swim anyway, but I do a very good bit of sinking. All of that after 10 years of lessons, in which I hated every single one.
Horse riding - try Riding for the Disabled. They can normally put a plan in place to make it very quick and quiet, which is probably what he needs? I love horses, but I have to work at my own pace, not be shouted at by instructor or made to do something too scary.
Otherwise, as the others say, find a hobby he likes. Don't worry about socialising stuff - just let him pick something that's right for him. We socialise differently anyway, playing next to someone, not often with them, and often love being online (like here) rather than face to face.
Ever thought of getting a dog, if the family circumstances allow it? Dog training classes are something he might like doing with the dog, and it's a great way to teach a child about respect and care and to meet other people in the park?
PS, watch out for any doctor that tells you it's "mild" autism. I know what they're trying to say, but Asperger syndrome etc isn't 'mild' if you have it, it just seems different from most lower-functioning autism because we learn to disguise the problems better. The problems are still there.