Personally I wouldn't worry
You said he is
"absolutely fine in shops, meeting people when out and about,visiting family,having family over to the house etc."
Seems to me he is getting all of the socialisation he wants/can cope with at the moment.
My elder child definately is on the Autistic Spectrum. When we first deregistered him (aged 13), he didn't want to go anywhere!
I remember offering to pay for any icecream he wanted, if he would just go and get it for himself. He shut the car door and said 'Don't need an ice cream that much"
He spent the whole of the first home ed meeting we went to, on my shoulder, saying "Can we go home now, let's go home now, can we go home now?"
Gradually slowly he came out of himself and began to mix.
At his own pace.
By the time he was 16, folks at home ed camps didn't realise he was anything to do with me as they never saw him with me.
By 17 he got a job by himself, took himself to FE college where he fitted in really well academically and socially.
He then chose to work and save until he was ready to go to Uni (so he doesn't come out with so much debt) He will be 23 when he finally gets there.
So it is later than everyone else, but so what?
Life I have learned, is not a race.
My two daughters have done things at a different pace again- and in some ways have raced past their older brother, but again in their own time.
Your son will get there in his own time-and be more confident for doing so than if he had been pushed IMO.
And not everyone needs big groups of people to sociaise with. Many children I know with ASD are much happier in a one to one/two situation.
Hell many adults are happier to do the same.
There is an interesting survey done to answer some of the accusations of the Badman review recommendations.
In doing so it looks at "as many real examples as possible of the type and frequency of community and social contact that our children actually have."
Not Hidden Home Educated
You might find it interesting.