I don't think either of us have any idea how "doctor contacts" are spread amongst state and private schools. It's likely that there are a lot of parents in state schools that have no useful contacts. And "contacts" is always cited as a reason to choose private school.
I agree it's likely that 18 million parents will have more contacts than 1 million, but it's no good to a state school student in Liverpool that there are loads of doctor parents in leafy home counties state schools. What's important is the number of contacts an individual might have.
Your DD might not have any contacts in her cohorts but private schools are very good at keeping in touch with alumni, and also with making connections across schools, so she is in a better place than an equivalent student at a rural state school.
Affluent parents also tend to make friends with other affluent people, so whilst there may be no actual doctor parents in the school, there is more of a good chance that a parent might be friends with one.
Or, put it another way. Do you really think it's more likely your DD will find a doctor contact if she went to a rural state school instead?