Assuming this machine is geared down and rotates at ~60 rev/minute or less, this will actually be pretty easy & economical. No fancy servos. encoders, PID loops or PLC required.
Mount a big slug of iron on the rotating frame.
Mount a magnetic proximity sensor on the base of the machine, where the slug will pass nearby with each rotation.
(using a cam and a switch will require much-more-careful alignment, and will wear out the switch relatively quickly)
(before making your permanent mounting bracket, make a temporary bracket so you can position it in the right place -- the place where the machine begins stopping so that it coasts to a stop where you want it to stop)
Connect the normally-closed (N.C.) contacts of the proximity sensor in parallel with the stop button. (it may be necessary to add a relay to invert the logic)
Now when you press & hold STOP, it will keep turning until it's in the desired position.
If you want it to respond to a momentary press of the STOP button, add a sealing relay which unseals immediately and turns off the green light to let you know that the machine has responded to the STOP request.
You could also add a time-delay relay so it automatically stops (in the right position) after an appropriate amount of tumbling.