It's definitely not the same. I still work in this field.
(Adam Kay describes the 'old school tie' system very well in his most recent book - not a particularly good read - but accurate.)
A child in my inner city schools (worked in 2) just doesn't have the same world view around them. It isn't just class sizes either. It's the total exhaustion of the teachers, the incredibly huge work load they have, behaviour in the classroom affecting the quality of teaching, the difficulty of expelling students (no issue there in private ed, they're just asked to leave).
Private ed kids have not just wealth but contacts. Private tutoring on top of private ed. Medical families tend to have children that go into medicine so those contacts are there, also knowledge of how the system works.
Although my DC is at state school, because I work in education, I understand how it works and I'm working very hard behind the scenes to advantage them. Many of DC's classmates don't have parents that understand this and as a result, their children aren't getting the support mine is (I'm happy to help if they want it, BUT it would be patronising of me to assume).
You're being deliberately obtuse if you try to deny the outright classism and lack of equality built in. All this initiative is doing is trying to expose state school students to a view of medicine as a possible career. Something that given many of those children's backgrounds, they may never have considered.