The playing field can never be level. Where I am the grammar schools are as good as the private schools, but they are only in places where property prices are prohibitively high (ofc)...but they are the only effective way out (Sajid Javid went to a comp, it is possible but it is very rare and requires support from family that isn't available in most of the UK). The way to move forward is to aggressively stream students (not necessarily by ability, I went to a school with a guy who was thick as pig shit but went into St. Andrews because he worked hard, took his A-levels three times iirc), improve teaching, and offer more opportunities in poor areas (maybe even through positive discrimination, i.e. offer kids from poor areas a chance in a decent school...the UK used to have state-funded places in private schools).
You are right though, it isn't only cultural. Part of the reason why this group is so far ahead is their education system is far better.