It's another of these things where there wont be an ideal system, even if you abolish private schools then there will be a natural geographical imbalance where people who can afford to live in affluent areas will benefit from the attached school which will be likely to have higher quality teaching (this sort of tiered effect can already be seen in some areas). Of course this is mainly a problem in cities which are perhaps compact enough that you could enforce schools to take in a certain portion of pupils from less affluent catchment areas or there might be some other clever way around this.

For me I grew up in an area which was remote enough to have one choice of school, then there's an extent where quality of the school is determined by how good the head teacher/staff are and will always vary up and down a bit. Think it worked well enough though there was a wide range of ability of kid there who have gone on to a wide range of different jobs/lifestyles, seems fair enough to me.