A true equality of education system would see all private schools and, in the British system, all grammar and high schools closed. There would have to be a rigidly maintained common syllabus across the entire education system, no child would be allowed to be taught any differently from another. All "home schooling" would have to be banned.
Unless you made it illegal to do so those parents with the financial means will hire private tutors and coaches for their children out of school hours. This would perpetuate the present system by another means if there was to be any selective tertiary education level. There is no way a non-selective entry system (first-come-first-taken or random) nor a 100% entry system to university could work and produce the quality of graduates.
A common education to 12 and then selection by academic achievement, regardless of social status, into academic and vocational streams would be even more divisive and very expensive. And still require the banning of private tutelage.
I think the biggest problem is not actually the entry system but the elitist "old boy" system where a mediocre graduate of a prestigious uni might be selected over a brilliant grad from a red brick. I think this is dying out, very slowly.
Also the sheer number of "lower class" unis now in the UK has produced a large number of "lower class" graduates that has made their degrees worth little more than a further education college diploma. Inflation in numbers can equal deflation of quality.
It's not what you know but what you can do in most fields of work. For all its problems the current system, in the UK, at least is hard to better, one hopes that employers will select by merit only in future - I think they are doing so.