I don't think the way education's so uneven across the country helps, and I'd say that's currently more of an issue than private schools alone in my opinion. It's much wider than private vs. state.
Where I live there's 2 private schools within a 45 minute drive. I counted to 15 state secondaries and then gave up. We have no grammars. There's no 11+, just standard academies. We're an area with high levels of poverty despite low costs of living.
I'm in the north east and those who want to escape their beginnings seem to end up moving away as we don't have the city jobs. You're looking at care, admin, NHS, recruitment, teaching, or manual jobs mostly for local employment. Those who move are then competing against people who are from areas that have grammar schools, so they had that chance even if their parents couldn't afford private. That automatically impacts on their chances within some areas of employment, as before they've even left school it had a part to play in their likelihood of achieving high exam results.
There's various debates around things like private tutoring for the 11+ meaning it isn't a level playing field (which needs sorting. 2 students in the same area, with the same levels of academic ability, should both have the same chances of entry but currently don't), but it's even less even if you've had no possibility of using it. The whole education system needs levelling. Private, grammar and state.
I'm on the fence about private schools, but banning them won't solve social inequality and issues within mobility if we still have a system which works with such a postcode lottery.