As things stand, public schools have charitable status, which, perhaps predictably, I think should be scrapped. By definition, many of the parents who send their children to them can afford generous contributions, and to pay for the sorts of extras that state schools need to provide (if they can afford to offer them).
State schools are underfunded, and often don't have the ability to make up shortfalls by asking parents to cough up. They are also more likely to need to 'level up' before they start teaching children who have not been socialised into familiarity with books, or other sorts of behaviour which makes teaching them possible. This situation is deeply divisive, however you cut it.
Comparisons with the 50s and 60s don't hold true. The demographics of the day meant that class sizes were larger, and expectations were lower, so what counted as a successful school would have been different, even if schools had had league tables, which of course they didn't. More able children were creamed off to grammar schools, which got better funding on the whole, and anyway the school leaving age was significantly lower than now, so less money was needed. Many people saw (and in some cases still see) education as existing simply to feed the job market, and back then there were jobs for semi-literate and largely ignorant people who just needed to pull levers or run errands. Those jobs don't exist in any numbers now - people need to have better qualifications in a wider range of subjects, so leaving without them is even more of a disadvantage than it used to be. In the past there were more opportunities for unqualified but able people to 'work their way up', either by learning on the job or by going to night classes or day release. Again, those opportunities barely exist now. It's just not comparing like with like.
Finally, poor old Diane Abbot is trotted out as a lazy example of 'Labour hypocrisy' in the same way as Mick Jagger is used to symbolise the need for means-testing pensioners. That is far more of an example of Groundhog Day than someone sticking to their viewpoint that we would be better as a nation if we allowed all citizens to reach their potential. It is perfectly possible for someone to disapprove of something being unfair, but refuse to sacrifice themselves or their children on the altar of 'parity'. Most of us probably spend far more than we really need to survive. Should we be disparaged as hypocrites for having more than one dress and taking holidays when we could pare things back and give the 'wasted' money to the poor? If we don't choose to do that, but instead opt to feed our children well, or to have a decent standard of living ourselves, does that mean that we don't care about the poor? I don't know what DA's circumstances were when her children were young (or whether her children would have needed security because of her job) but if she lived somewhere where the local school was poor, should she have condemned them to that while she worked to improve education standards for everyone?