I have a dd1 in Y6 and ds in Y2, so fwiw these are our experiences (inner-London primary, but seriously considered various other options over the years).
You can be lucky with a primary school and get one which is (a) a good school with experienced and able teachers and (b) has a majority of children of average ability or above from supportive homes. In this case there would be no need to look elsewhere, unless you felt strongly about some of the extras that private schools provide (more sport, after-school activities, posher facilities, social glitz).
BUT in many cases the reality will fall short of the ideal. Many schools (particularly in London) find it hard to recruit and retain good teachers. Many are also struggling with large numbers of children from difficult and disrupted backgrounds. A good school will be able to cope with this and still provide the children with a stable environment and decent education. (This is the case in my children's school).
However, the class sizes will be bigger, and there will inevitably be less attention paid to each individual child, so some children's needs may not be fully met. Teachers also have very little flexibility to vary the curriculum, and much of it is couched in an off-putting 'edu-speak' that private schools can bypass.
Even if your child is in the ability range that is being targeted by the teaching there is still likely to be less 'polish' -- handwriting is often noticeably poorer in state school children, for example.
On the upside, many of the private schools (particularly in London) can be very pushy and a bit precious and competitive.
In the end it comes down to the relative merits of particular schools and individual children. It's not the place for political rhetoric, though sadly this is often how it turns out.