My Dad was in the navy and they used to pay for children to attend private schools (maybe still do) so I went to boarding school from 11. I wanted to go (too much Malory Towers 😃) and loved it. I was very academically able anyway - it was a selective entry school - and came out with pretty good O Levels. But I wanted to leave and experience the world a bit so did A Levels at the local tech - so maybe didn't get all the benefits of private school that I might have if I had stayed longer. I didn't go to uni as I had met a local boy and wanted to work. Had I stayed I would have gone on to uni and who knows......
But I did do pretty well in the end - got my degree-equivalent qualification through work and had a fulfilling and successful career.
DSis was three years younger and went to the local comp, as we had stopped moving around so much by then. It didn't have a great reputation and presumably had distractions etc, but she did really well, and has done fantastically career-wise ever since. She did go to uni, and then got a masters.
So I think it's really hard to know. You might do better at private if you aren't that engaged at school, or need motivation. But if you are keen you should do well wherever you go to school. Even the lowest achieving schools still have some great kids and some great teachers!
I think the other differences are things like - making useful contacts, having great sports facilities, maybe not picking up bad attitudes or mixing with the wrong crowd if you're easily led, etc? (Although people always say that our local private school has just as many dodgy kids as the comprehensive - but theirs buy cocaine whereas ours smoke fags behind the bike shed.....😂.)
Sorry - no helpful advice. I would never pay for it at primary as I think the benefits are more at secondary. But every situation is different.......