You pretty much summed it up above. The main reason for contact between the NY families and the Outfit in the past was 3 of the "Big 4" unions and open territories like Vegas. Those unions aren't controlled on a national level anymore and there's little mob presence out west so there's not much reason for interaction between New York and Chicago anymore.

As to the original question, New York had influence over who got made in New Jersey and Philadelphia. They also forbade Pittsburgh from making any new members after a certain point and Cleveland had to get permission to make new guys.