I think the key here is that there is such a spectrum of both state and private schools, that the answer might be different for everyone - even if fees weren't a consideration.
We live in North London. My dd is still 5, youngest in her year 1 class. We were allocated a place at our 3rd choice (but closest) school in reception. Failing school 5 years ago, now Ofsted good. Single form entry of 30, beautiful playing field, fantastic pastoral care, a very wide variety of after school activities despite its small size. Dd is happy and confident and has lots of local friends.
HOWEVER, we are going to move her to an independent school for the start of year 3 because:
We prefer a more traditional focus on academics and a bit more of a 'push' (she is bright based on end of year 1 NC levels). At her current school no handwriting taught and have only just started learning simple spellings. 1 reading book a week. We feel that we have to work with her if she is not to just coast. Although the school is lovely, their academic aspirations are mediocre.
We prefer a larger school with multi-form entry.
We prefer a school with an even wider range of after-school activities all carried out on site (we both work full time) and largely to a higher standard (she is very sporty).
We want a school that will take her through to 18. We live in a grammar school area but they are super-selective. She might get in but we can make a decision about whether to sit 11+ later.
We like the emphasis on an enhanced/broadened curriculum. Lots of lovely new experiences and days out etc. Not just about trying to 'accelerate' kids academically.
However, she will have further to travel, will have to make new friends and we will have a lot less income. It's not better in every way at all (and this school is highly ranked in league tables), but it's just better in the ways that are important to us at the moment.