Does it matter how many contacts a student has with the school?
I think it depends on the college and the kinds/numbers of contacts in question. Having an aunt who went to the college 20 years ago is different than your dad playing golf with the provost.
My own children have (literally) 14 relatives who graduated from a certain selective university, but it seems that they will have to make their own way; legacy SAT scores are actually trending higher at that particular university than scores of the rest of the acceptees, which means it's actually tougher to be admitted. (I will clarify that I don't think there is a bias against legacies per se, but that the school is simply reaching into different demographic segments, and with more fervor).
Reach out to all your contacts, though. Even if a contact cannot provide direct assistance/influence, they often can help indirectly, by letting you know more about the institution and the best practices for admittance.