"Couldn't more energy be spent on improving the poorer performing schools"
How? With a tiny, tiny number of exceptions, you can work out how well a school will do from the proportion of free school meals. Arguments about exceptional schools in deprived areas are about added value, and just because a school does very well at getting people with very difficult backgrounds through a few GCSEs says nothing about their ability to get middle-class children good A Levels (and, of course, vice versa). As to why that is, it's a political and moral third rail, but the connection between parental income and the outcomes in schools is uncontestable.
To put some other canards to bed: you can't "ban private education" because (a) it would of course involve increasing the number of state places that are funded by the number of people in private education and (b) unless you're going to simultaneously make home education illegal, you'll also have to stop co-operatives of parents running informal schools, which would almost certainly breach the ECHR.
Labour accept the link between deprivation and poor educational outcomes, because that was at the root of Sure Start. That SS was colonised by the middle-classes says interesting things about relative enthusiasm for education, but little else.
Short of removing children from home at birth and raising them on Kibbutzes, the inequality of outcome is almost inevitable, because affluent, educated parents can in general both (a) assist their children and provide additional opportunity outside school and (b) provide a positive role-model for the advantages of education. How you overcome that is the challenge to the left; how you ignore it is the challenge to the right.
But until people square up to the fact that a major cause of good and bad schools is good and bad parenting, and address how you improve the quality of parenting which does, for the purposes at hand, appear to be linked to broad measures of income, this is all a sterile debate. Children arrive at school at five already segregated by parental influence, and until you address that, worrying about admission at 11 is futile.