IMO majority of people who choose private v state do so for more reasons then just reading ability(reading level not being the only part of child education that counts)
Things such a wrap around care, facilities of the school (sport, music, art, drama) small classes, attention to detail and so on.
I don't think children in private schools are ahead of their peers in state schools as per say, but there have more opportunities, often good prep. and independent schools feed into excellent secondaries.
IME, DD is in local independent school with excellent record and reputation, and she is there for lot of reasons, pastoral care and gentle approach being one of them, not the academic excellence alone.
She is excellent swimmer, in the class of only ten, they all read pretty much the same level, but they are all very articulate, very able writers, and few of them excellent at spelling(way ahead of others).
In addition to beautiful food served at lunchtime, cooked breakfast if children arrive early, beautiful grounds, tennis courts and in general whole look and feel of the place.
DD has a cousin of similar age in state school, Y1 who is currently on higher level of reading, bright little thing one of 30 in the class, and struggled great deal last year due to disruptions of a such a large class, in her mother words:
dd vocabulary is lucking, her comprehension is not great, and I wish they would slow down with reading levels and pay bit more attention to comprehension.
IMO different children have different needs, there in no wright or wrong, perhaps OP's question should have been phrased differently, to get an accurate answer.