I'm going to start at the beginning though you seem to have covered a little of this.
Does it cycle the ejector arm and bale? As if it was harvesting a load and checking to see if the bin was full?
It probably looks like this:?
You've mentioned that you saw the gears. Unless it has been changed it should have the jumper holes, they did from the factory. I've actually never seen an older one that didn't and some still do. If you pull the white front cover off are there 2 gears there? A little and a big?
If so, the little gear may have a slot in the middle. Stick a slothead screwdriver in it and turn gently counter clockwise a few degrees. It should proceed into a harvest cycle from there. If not turn it a little more. It will not try to fill until the end of the cycle.
The next computer controlled generation did not have any way to jump the thermostat but would initiate harvest if the bale arm was lifted and returned fully 5-6 times in rapid succession. Try that.
You may have to hold the door switch in to power the IM.
Beneath that white cover should be something like this. This is a whirlpool version but GE were similar. Neither manufactured their own. Yours should have a small gear on top left of the large one.
In this one you cannot turn the gear manually but you can jump the thermostat. It's the holes labeled T and H. You'll need a fairly thin long wire to reach the internal contacts, and it must be insulated where you hold it or you may get shocked. When you make the circuit it'll begin to turn the ejector. Hold that jumper in until you hear a click, after that you can remove it. It'll proceed from there. You may have to hold the door switch in for this too.
As a last resort you can forcefully lift the ejector fingers to rotate the head and sometimes get it started that way.
Finally is the freezer actually cold enough? The thermostat will not initiate a cycle until 15 degrees or lower. If your ice cream isn't firm it may not be cold enough. Also ejector fingers can get stuck or jammed.
If you can get it to try to harvest you can use your meter to see if it is supplying 120v to the valve at the end of the cycle. Just probe the connector at the valve. A final thought; is the freezer door switch functional? Turns light on and off, fan also?
Try all that applies and if not post a pic of what is beneath that front cover. or at least the whole unit.
Good Luck