Sorry, haven't rtft and am not sure if you're looking at primaries or secondaries. What I've written applies to England only.
There are no restrictions on class sizes after the first two years (R, Y1).
State school classes do tend to be larger than private school ones but all schools will want to fill the rooms they have. The economics of the school sector (any school) are very hard right now. Energy costs and labour costs are sky high.
State schools are often dilapidated unless parents have done extensive fundraising. That was a deliberate govt policy choice when they cancelled Building Schools for the Future in 2010. Now there is a huge backlog maintenance problem.
Private schools are generally very well maintained as prospective parents judge them on that.
Very little difference between teachers in both (I've worked in both). I actually prefer larger classes for A-level (20 ish is my ideal) although marking can be a beast once mocks roll around.
Generally, however, academic standards have improved markedly in the state sector but that's come at a cost of breadth.
It is fair to say that (classical) music, drama, sports, classics, languages are better served in private. You need specialist equipment and teaching to do a lot of that well. Independent schools do share their facilities. But may stop now due to the toxic public discourse following the VAT changes.
But you can pick an area with a good youth music/drama organisation or the type of sport that you want if that's important to you. Still charities doing that stuff well.