Around here (central Indiana) I don't see very many big cue balls, but the heavy or mud ball is prevalent. All the league players have their own green or purple Aramith balls that they carry with them to swap out with the heavy cue balls at the bars.
What I find is that the leagues are generally not in the bars today that use the big ball. In my area, all the areas that have the big ball... no APA or BCAPL and associated players are walking in for a tournament or league !
In the past, when leagues were relatively new, we'd play in every imaginable table and every imaginable ball. I played in a social club that had a coin op 9' furniture table (Sterling Table)with slow cloth and with a big ball. Total home team advantage!
So I think the original question from Bob is a bit two-fold:
Are there places that still use the big ball (Yes, a zillion across the country).
Are leagues/tournaments/semi-serious play associated with big ball play (Substantially lower percentage today than twenty years ago)
Freddie <~~~ has a big ball