"He's quite a shy guy, so it's quite a big thing for him to even suggest doing something which will take him outside of his comfort zone. Which is one of the reasons I'm broadly in favour. Perhpas he feels the time has come to challenge himself?"
That would be an important reason for supporting his choice to move, imo. Assuming he hasn't got his heart set on one particular highly specific and competitive university course, acquiring a good level of social and intellectual confidence is likely to be of longer-lasting benefit to him both at university and beyond than a few UCAS points.
But it does depend on what he wants to study - if he's hellbent on Oxbridge, or becoming a barrister/city solicitor, going to LSE or Imperial, or reading English at Warwick or whatever, then he needs to follow whichever path is likely to maximise his A-level grades. If he's an all-round above-average bright boy who will get good but not genius-standard A-levels, get into a good university but is not likely to be a future CAmbridge First (this is my ds too, btw) then I'd be encouraging him to branch out if that's what he felt was best for him.
To put it another way: there is a big learning curve between a small, protected private school and university, and an even bigger one between said small school and the big wide world. If he is a slightly shy, tentative boy then he will have to climb that curve at some point. There are arguments for doing that at 6th form, and counter-arguments for waiting till university. But if he wants to move, and there are no particular deal-breaking reasons not to, then I think I'd be cautiously supportive.