I went private, my parents were teachers so saved every penny to send myself and my sibling. I was always academic, so I think I'd have done well anywhere but I was incredibly shy and nervous at 11. My school built up my confidence and helped me to feel seen in a way my primary (90 kids per year vs 40 at private secondary) hadn't. I was surrounded by families who got on in life, not mega flash but no one was failing. It sort of set this expectation that anyone could do if you set your mind to it, because none of those parents were mega geniuses that I can tell. I went on to found my company, have more than 20 employees, I genuinely think if I'd gone to the state school I'd have gotten a decent job, but the idea that I could do more than that would never have really entered my mind. Likewise my brother was very quiet at 11, he still is as an adult, but he's also the most self-assured person I know. Kind, quiet but he's very secure in himself. We weren't at the same private school, he picked a much sportier one and mine was single sex and more academic, but for both of us I'd say it helped our grades, but not hugely, but more over it's set both of us up very well for the rest of life. Personally my one regret was going to a single sex school, but at the time there wasn't a co-ed private within 50 miles, by brother was the first year at his school that took both.
Two of my friends are secondary teachers and both very strongly advised private if we could at all afford it. What struck me more than anything was one of them saying 'I see each child an hour a week, that's more than 700 a week, unless your child is genuinely gifted (in my subject) or a real troublemaker, I won't even know their name until GCSE'. In comparison we get phone calls or emails a couple of times a week from our daughter's school to update us on anything we need to know specific to her (she has some issues which require extra support) as well as obviuosly all the extra-curricular etc that state just can't match. I do also like that they don't have to follow the national curriculum, which in parts is antiquated and irrelevant, so the education can be more relevant.
Our daughter has a strong interest in IT, which is the area my husband also works in. We went around 3 privates and 2 state secondaries. He spent a long time talking to the IT departments in each, 2 of the private schools taught to a standard that he could have any of their pupils at 16 walking into a 45k trainee role, the other private much more followed the standard state curriculum which he felt was not close enough to what was needed in the workplace now without requring a degree to get there. It's not the fault of the school, but the curriculum that is set for them and not adaptive enough to changes in tech. This in part made our decision because what we spend on fees in 7 years could be easily the difference in earning capacity in the following 5 years.
With all that said I don't think all private is created equal, some isn't going to be worth the money, and some state ed is worth going to (but our town secondary is supposedly in the top 200 in the country and yet no one has a good word to say about it). Speak to the parents in your area about what they think of your schools in particular, league tables are much, much less important than the outlook they each have to nurturing your child.