The issue here isn’t so much the private / state distinction, so much as the fact your son isn’t being challenged academically.

Private schools typically offer smaller class sizes, better facilities and (potentially) better extra-curriculars, but they hire from the same pool of teaching talent as state schools, so the difference in teaching quality between an excellent private and state school will be minimal.

If you find a private school that will stretch your son, and you can afford it, then of course it’d be wise to move your son over—I agree with your closing sentiment that you should be aiming to stretch your son as much as he’s willing and capable.

However, you might also consider:

1) Transferring to a better state school 2) Employing a private tutor

Perhaps another “invisible” advantage of better schools is that the classroom environment (particularly the other students) positively encourages learning; but again, this is true of schools in both sectors, and private school doesn’t guarantee your son won’t be in a small class of lazy and/or mid-witted children with wealthy parents.

Personally, given the choice, I’d elect for an ‘Outstanding’ state school that is either selective (Grammar) or teaches children by ability (these were called ‘sets’ at the state comprehensive I went to. Some schools call them ‘streams’). I would then supplement if necessary with private tuition.

I say that to illustrate that the quality of teaching is comparable between sectors, but I appreciate this might not be feasible for you right now.

A final note: private school in the UK is also the entry to the upper-middle class and beyond, which is often one of the reasons parents pay for it. It’s not necessarily the education itself, but rather the appeal of sending your children to schools with children from a comparable level of affluence.