Not quite true. The first cellular service in the US was launched by just prior to the January 1, 1984 divestiture of the Regional Bell Operating Companies (RBOCs - the local service providers of telephone service) from AT&T - the long distance provider. The March 6, 1983 launch was by the future RBOC which would become Ameritech. So while technically not AT&T, it was the spawn of AT&T.
And one certainly didn't get cellular service back then because you were tired of being overcharged by AT&T. A normal cellular bill back then was way over $200/month - in 1986 dollars. And that got you maybe 60-70 minutes of air time, and didn't include long distance either.
Back in the day the FCC, through spectrum allocations, mandated that each major future cellular (they weren't called wireless back then) market would be divided into two: the non-incumbent "A" service provider (e.g., like a Cellular One in the NYC metro area) and the incumbent "B" service provider (like NYNEX in this case). So in lots of markets wireless startups were funded by investors who'd make a killing. But in other markets, the non-incumbent carrier was simply another RBOC which was entering into a new territory.