It really is horses for courses.
Both mine went to a reasonable local comp, got excellent academic results, did music lessons etc, lots of extra curricular, sport available (though both focused on sports outside school). Neither had tutors.
We’ve always taken the view that much of education isn’t actually what’s learnt at school. The money we’ve saved has gone towards books, holidays focused on nature, art, history and so on, funding one DS’s (expensive) sport to international level, helping both at university, funding the other DS’s Master’s year and so on.
One DS spent the gap year between Masters and (funded) Doctorate tutoring and cover supervising/teaching at a very tough comp. Most of his tutees were at private schools with a few at grammars. As a young cover supervisor he tended to end with the less productive classes, let’s say, but also saw that there were bright, hard working kids who did go on to get decent results and go to Oxbridge/Russell Group etc universities. These tended to be the ones with the supportive parents, unsurprisingly.
Obviously, there aren’t overall conclusions to be drawn from this, just that it’s not black and white. Some children will need the extra support they might (but don’t necessarily always) get at private schools. Many won’t need that extra support and can get the extra curriculars from state schools or elsewhere.