I like the idea of the plug interface so you can revert to the original manual controller. What happens if you continue pulsing the step button when Gabe has already reached the top? Will it break or is there some kind of clutch that disengages further motion? Not having one in front of me, this was my motivation to install a end-of-travel detector so that the climbing stops at the top rather than depending on a fixed number of pulses.
In any event, you should be able to find a 1 RPM DC gear motor I'd guess in the $10-15 range surplus. Since you have virtually zero load torque (some cams which hit leaf switches), a higher RPM gear motor could be driven at a lower voltage to reach 1 RPM and still work smoothly...for example, a 5 RPM motor should be no problem.
If going the motor route, you might consider stacked cams with each concentric cam activating one function. One cam could be the motor power control which closes its switch for 59 seconds or so. To start the animation you press an electrical pushbutton which is in parallel to this master switch. The motor starts turning and within 1 sec the cam switch takes over and continues to apply motor power for 59 seconds. Then the 1 sec notch in the cam comes around and the motor stops until the next manual activation. A 2nd cam would have the dozens of nibs for stepping. A 3rd cam would have one long nib to turn the lights on and off....presumably leaving them on when the motor is stopped.
I suppose your method is the most straightforward. If I were doing it I'd probably use a programmed controller chip or module (like the Arduino) or perhaps a discrete controller circuit using counter/timer chips. In any case, I'd experiment with the ratcheting mechanism to see what kind of DC pulses would work since that would allow pulsing to be done with transistors rather than relays or triacs. Likewise, I'd drive the lamps with DC which simplifies control from logic circuits.