The movement of each mixer is timed with rotation of timing cams on the control unit.
Mixer #1 is the exception. Each mixer portions odds and features through the game. Each one does something slightly different.
Basic idea is that for each movement of the start relay, the spotting disc releases and lands on a random spot. The machine will allow or not allow features or odds step up (if the river on the spotting disc is hot), through the various mixers (depending on rivet), and the reflex.
In certain instances, the reflex is bypassed.
So you have at a minimum, a 1:3 chance of a step up, but usually much less. The real magic is that step up can also be prevented based on what is currently present on the game! A good chunk of complexity is present in the mixers and some of the real genius that went into the mechanics of these games.
The numbers indicate position if you have the diagram of the mixers (of which Phil Hooper has documented quite a few), and for each position you have like a 1:4 pass through for certain features run through that mixer via the spotting disc.
Mixer #1 has a guaranteed 1 step rotation and must move to allow anything to happen. That's most likely the biggest problem with your step ups. Over time, or with moving the game, mixer cam riders might get stuck open. Get on a step stool and examine from the top. Look at the bottom of each cam and you will see the latch/rider. It should be sitting in one of the divots. You can see mixer #1 (which has a second hold rider) is stuck in the down position.
Free it up and I bet your game will start stepping. Check for packed old grease. Also typical.