What would be a way to acquire this sound without actually building or finding metal gears that turn?
You need two components - metal clangs for gear hitting gear and grinding sound for the shaft rotating. Record lots of metal clangs - you need many variations and often finding the sound that matches perfectly takes some time. Experiment with samplerate of existing sounds. Grinding sound should match to material shaft is connected to, but if it's unknown just use some metal grinding sound.
Play metal clangs in a sequence and adjust the gap between each clang according to gear rotation speed. With low speeds decreasing volume is usually a good idea too and you could try lowpass filtering the clangs too. With grinding sound adjusting only volume according to rotation speed is often enough, but sometimes minor samplerate modulation can be used.
Optimal would be building an instrument with sampler that supports playing a sequence of sounds and modulating trigger rate. Can't recommend any, since I use game audio middleware for that kind of purposes.
I've done lots of gear sounds and this is how I usually do it.
I would add, for grinding to a halt: the shaft component of the sound might screech, groan, shudder, go down in pitch, or otherwise respond to the changing force, while the clanks of the teeth will slow down and stop.
Are the gears visible on stage? This is something you'll want to set aside tech time to practice the sync if so.
Good luck, sounds like fun!
There is one possibility for a field recording that, if successful, would be very easy. Most overhead roll-up garage doors (on commercial warehouses, not residential) have a manual override mechanism: a chain that runs through several gears and pulleys. Affix a contact mic (or two) to the chain and hoist away. Easy to control rate, and the door panel itself usually squeaks and grinds as it lowers. Very complex and rich soundscape.