... and it's not a tachogenerator either
Given the avionics connection, my bet would be on a synchro/resolver too. In the old days the rotor feed would have been via slip-rings but obviously things have moved on with the electromagnetic coupling. Excitation would be 400Hz (or possibly higher for this setup).
In terms of decoding, I'm not sure how much effort you might want to put into actually trying to use this bit (unless for pure interest) as the will be both amplitude and phase to decode. Given that you have an exposed shaft available you could retro-fit whatever you want, be it slotted disk and led/opto, magnet and hall sensors / reed switch or nothing at all depending on what you decide to use it for.
I'm a bit puzzled that you didn't see any increase in mechanical resistance when you shorted it. It might be worth taking a look at the state of the brushes, or it might just be the characteristics / operating voltage. If it had been me, I'd have had put a few volts on it by now!
One thing's for sure your garage door opener hypothesis is out of the window by now - even if it might actually make rather a good one