does a speed controller act as an inverter?
No, speed controllers can work several different ways depending on the motor. Also, finding a motor and speed controller that will run on 120 VAC with a line cord shouldn't be a problem. I saw back there you mentioned I believe 30 to 100 RPM? What was that about? Not a problem but thought you wanted 1 RPM? You can find small very light application 1 RPM AC synchronous gear motors pretty cheap. They won't drive much.
Ron