Profaci was more in line with a Genovese than a Gambino. He secured immense power for his family whilst throwing their lives away if even slightly provoked.

The american mafia yahoo group has a bunch of old archives regarding mobsters from the 20's and 30's. There was a thread speculating that there was a Brooklyn boss whose family preceded Profaci, thus explaining his almost "instant" rise to power (it wasn't so instant.) I couldn't find the thread but I believe this is the website.

https://groups.yahoo.com/neo/groups/americanmafia/conversations/messages/263

^ A separate conversation regarding Profaci

If anyone can dig it up there was a lot there.