Almost 50 years ago, where Atlantis is now was our water sheltered skiing beach on the East end of Hog (now Paradise) Island. We launched from a mooring that we waded out to from the Ft Montague area. But all that's in the past.
I'm assuming the remaining charter boats are still in Nassau harbor on the Nassau side of the bridge (they were in 1979). There was next to nothing on Paradise Island besides the hotels and beaches and beach equipment. The same is true for the West End hotels - there are no good anchorages or mooring spots on the West end of New Providence once you get outside the harbor.
Nassau would be a hard sell to me to run a charter business out of. Expensive to have boats there, and a very uncertain business model. Also, not many places to go within 5-10 miles of the harbor that haven't been filled in with hotels. Abacos, Eluthera, Exumas all have lots of places to visit for charter boats - that's why the businesses are there.
Fred W