I think it's ridiculous to say it doesn't help. It isn't everything, but of course it's going to help.
Small classes, more individual attention on its own will help. Not having the set of disinterested learners disrupting a lesson will help considerably too. Yes, private schools can have that as an issue, but typically much less.
My df went to a secondary modern. He got A-levels, the first person from his school to get them. His A-level maths teacher took the exam at the same time as him and got a lower grade. he could have come out with a string of As as very intelligent, very hard working and motivated.