I have some experience of both sectors at primary level with different sorts of children, all in pretty high achieving schools, so I think comparing fairly...
At primary level my observations are these:
1 - your child will not fall behind at a decent primary (state or independent) IF you are on top of their education all the time. By this I don't just mean ensuring they complete their homework, but I mean ensuring they are following the curriculum, that there are no gaps, extending them where the class is slowing down, topping them up in areas they are struggling, and doing plenty of extracurricular, including museums, etc...do not underestimate the effect of a bad year with a bad teacher or lots of supplies, this will be a huge commitment in terms of catching up and do not rely on a teacher saying "your child is fine" because it might be true but it might be meaningless in the context of their progress. Follow your gut feeling.This is true in any sector you choose.
In a same vein do not rely on NC levels to determine whether a child is bright or has made progress, but rely on your own instinct. Often children will be not taught at the next NC level for lots of reasons, for example because they are at quite a high level already or they have one single gap which needs filling, or they are not clicking with a particular teacher, or there is some other problem lurking which has not been identified.
2 - a lot of the success stories will also depend on the cohort your child will be in, at the same state primary one of my children was in a very low achieving cohort and one in a relatively high achieving one, this meant that the latter was taught a lot more and was much less of a worry. Although most schools do differentiate, most of the teaching is still done as a class in both sectors, so the median level of the class is very important.
We found that for example, in Y4 my DS1's class (state) was being taught at L4, the prep was teaching at L4 (so very similar), BUT my other child's cohort was being taught at L3 still, because most of the class still hadn't mastered L3. It was much harder to top up the latter and my child was very unhappy, so we had no choice but to move her.
Also depending on the child, motivation can become low if the teacher doesn't recognise that they need to be kept stimulated. This is very child dependent but in most cases a very bright child is not a good match for a low achieving cohort, especially if it is coupled with disruption in the class or mediocre teaching.
Finally, for me the sign of a really good teacher/school is when a child comes home fired up about their learning, opening books, revising timetables and really getting stuck in their homework. I have had this in both sectors with different teachers and same children, so although I would have loved to believe that it is all down to the individual child, it is often a sign of how inspiring that teacher is to your child.
Hope these considerations help a bit.