Servos are not like regular DC motors. First, typical servos are not used for continuous rotation... Second, you can't just apply DC voltage to it, you need PWM(pulse-width-modulation) circuitry to make it move. The strength of the servo is determined by its torque: the amount of force ( measured in oz) needed to move the shaft from its current position.
Even considering the motor you have used to spin the platter and the record on top of it, it stil might not be enough to use it in a cleaning machine. One thing I learned from my youth days, when I used to build R/C battle bots, is that no matter how you over-spec the motor, you always wish it had more power...
If you want a DC motor for DIY cleaning machine, you might want to looks at auto motors: the ones used in windshield wipers or to open/close windows... These motors are strong, hermeticlaay sealed, come with reduction gear, use 12V, so building PSU is not a problem, and could be bought dirt cheap at the car junk yards...
Cheers, Paul