I think it all depends on the particular ethos and expectations of the schools you are talking about, as well as the individual child's personality, interests and ability.
FWIW my (state-educated) children learnt to read significantly earlier than their (privately-educated) cousins, and are still ahead in most respects. Obviously you could argue that they would be still further ahead if they'd been taught in a more academically demanding environment, but I don't think knowing 5 more tables at the age of 8 translates into better A-level results or a more successful career.
In the end, the primary curriculum at infant level involves teaching children to read, write and do simple arithmetic, as well as the all-important non-academic stuff -- taking turns, listening when the teacher talks, persevering when you find something difficult. As a bonus, you can add in some art, some nice PE, some music and a few fun school trips and activities. Any decent school can provide that. You can do it yourself if you have a mind to.
Find a school where you like the vibe, like the head, and like the look of the children that come out at the top end of the school. The state-private thing is a red herring.