And "contacts" is always cited as a reason to choose private school
I disagree as I think the main reason is the education itself, help for individual needs in smaller classes with more teachers, provision of sports, music and drama, a presumption that most there will do their best and be aspirational, interested in their work and be willing to work hard. Some are very selective so the point is they are like a grammar school.
Helpful contacts come from a milieu where there are professional parents, whether their children are at state or private schools, and a headteacher who is good at calling on their help.
Private schools are now so expensive that fewer people can afford to send their children to them, especially if they have more than one child. There are plenty of state schools with children who are certainly privileged relative to children in poor disadvantaged areas.
The NE in particular has many very poor people. The exam results are dire compared to the SE, though London schools seem to do better even for the poor and disadvantaged, and the SE has grammar schools for those who can afford to live where they are. The poor NE children in difficult areas will most likely not get the chance to go to medical school and they could be held back from that long before the work experience stage.
The best chance for a poor but brilliant student who is living in a poor area would be for their head teacher to contact a private school and get them a bursary there (if they would like to go), or failing that, a teacher at their school who takes them under their wing and mentors them.
Perhaps regular general mentoring and encouragement from willing professionals would really help a lot too.