Bob, even if they are using just one cycle, it still has to self advance at some point, wouldn't it?
Unless you're saying there is a continuous timer cycle for the wash actions AND
another timer that only allows the machine to run for a certain period of time once the coins are inserted with that time being the amount of time it takes for the cycle timer to revolve through one cycle.
I've only helped a relative to service a 'front servicable' mid 70s GE FF coin-op, similar to the one in the photo, once in 1981 but we had to replace the motor and clutch. Didn't do anything with the timer.
When one looks at where the console lights are on the non-coin-op model vs. the home model, it looks like they're in the same place.
I'm thinking they deleted some of the features for the coin-op, like the suds return and other cycles, then wired it to only use the one cycle. Then when you put the coins in it auto advances to that cycle and completes.