I have worked in schools and have lots of friends teaching in different schools. One of the main differences re public v private is class size. In a private school, there were 12 kids in a class, most if not all, had private tutors on top so the kids understand very quickly what is being taught to them and you can get through the lesson much quicker. There is less admin for a smaller class (planning, marking, reports etc) so the teaching staff could do it all properly.
In a state school there were 30 kids in the class, 14 of whom had SEN. Most of the lessons were spent on crowd control so it was impossible to get through the material at the same pace at all. The admin side was insane so most teachers would cut corners here and there. This is more to do with class size than teaching ability, so perhaps you could look at some nearby village schools and see if there is one with smaller class sizes.
The other main difference of course is resources. It is much easier to have the latest tools and tech and resources and extracurricular provision when the school has £££.
One con of private primary is that there is a big emphasis on entry exams for secondary schools. Secondary schools tend to be selective, whereas primary are not. Lots of parents can get obsessed with their child getting into a particular secondary school, but some kids (however much money you throw at them) aren't academically bright and will not get in. The kids feel the pressure. The teachers are also under pressure to make sure the kids pass entrance exams so there are a LOT of practice papers and teaching to pass exams whereas in state schools, it can be a bit more child-led.
You learn more at school than just your lessons though, you are also learning how to navigate socially, and in private schools, you may not have the same exposure to social diversity. Ultimately, I know plenty of privately educated people in very normal office jobs and plenty of state educated people you are high flying in their career.
Maybe look at village schools and explore private music/language lessons if you want to supplement her education.