To me the Core are Syracuse, Connecticut, Georgetown, Providence, Seton Hall, Villanova, St. John's, and Boston College. Out of that core only Syracuse and Boston College played FBS football for most of the life of the Big East. UConn added it about 10 years ago.
20 years ago, the Big East bolted on Virginia Tech, Miami, West Virginia, Rutgers, Pittsburgh, and Temple. And the core was really selective with whom they allowed to play ball so to speak in basketball. Basketball was their main mission. I can remember Virginia Tech bolted on for a long time while playing basketball in the Atlantic 10. Temple was bolted on too. I'm not sure about Miami. Don't remember.
My original point was that the Big East was a basketball conference trying to accomodate football on the side. On of the benefits of the American is that everyone is on the same page.