Well, the timer cams index does allow for movement, and the mixer latch allows for the light show when you press the button.
Check control unit cam 3A - that switch allows for basic gameplay to function. If it's opening and closing as the control unit rotates, then it's probably not that.
My first suspicion is that the control unit motor needs oiling (there are two brass filling tubes that stick out from the motor - use 3-in-1 motor oil in the blue_ can. Not 3-in-1 in the red can. Put a few drops in each tube, then turn the game on and let the motor run for a few minutes. Then do so again. If the oil comes out the bottom, then move on.
Next, check the edges of your clutch washers - there's one set in between each portion of the control unit. In Showtime, there should be two sets.
The timer cams are the ones in question - if they are sticking, then the motor won't be able to finish rotation, which keeps the mixer latch active.
To add oil to these, I will /carefully/ use a flathead screwdriver and pry the piston-type thing from the washer. Then, I'll use a syringe to inject neatsfoot oil (comes in a cylindrical container) onto the clutch washer directly. Wipe off excess that drips with a paper towel. Then rotate the control unit (you can do so by hand - and you'll want to do this with the game off). Then do the other side. If this one set needed it, the others may as well. There are also clutches on the mixer unit at the top of the back door that separate the mixers from the spotting disc.
If the motor is laboring, then your motor needs attention. If it's not laboring, then it's probably a clutch-related issue. A little oil wouldn't hurt the motor, regardless. The mixer motor also has filling tubes.
Remember: 3-in-1 motor oil in the blue can for the motor filling tubes, neatsfoot oil for the clutch washers.