You can't generalise about all independent schools v all state schools - it really does depend on the individual schools and what they offer, and whether that matches the needs/talents/interests of your child. You need to visit any possible schools, ask lots of questions, do your research...
But if your DC is already established at a school (what year group?) and is happy, achieving well, has a good friendship group etc, then I would say the default position would be to stay put. Maybe rethink for sixth form, depending on what options there are at different schools (eg my nephew moved from private to state for 6th form, as the nearby 6th form college offered some A-level options he couldn't do at the private school - it can work both ways).
I'm happy with DD3's school generally but she's in year 1 and I'm not happy that the teacher seems to want to keep the children working at the same pace as others in their groups. Also that the teacher has listened to her read only once last half term (and a parent helper once). I did ask about reading at the last parent's evening and the teacher said she's been busy with performances, etc. I thought well ok it's only a few weeks into the year but nothing has changed. I read on another thread about a 6 year old getting a scholarship and it just got me wondering.
BUT DD2 who is in year 8 is likely to get mostly (if not all) A*s in her GCSEs and I'm very happy with the school. DD3 is likely to go onto the same secondary school and the only criteria we fit is that she attends her present primary school. So if I move her to a different school then we lose our link.
She is obviously a bright little girl, I would carry on with what your doing at home, don't worry about how often the teachers are reading with her, as long as she is getting plenty of that at home then 10 mins of guided reading once in a blue moon with someone at school isn't really going benfit her one way or the other - what is the teacher she's likely to have next year like? Maybe it's too early this year to judge the current teacher yet? It's doesn't sound like a bad school so dunno if one not so spendid experience this year is enough to warrent a move. You'll do whatever you think is right by her anyway x