Can you describe the gears and how they drive the wheels? I can't find a diagram either. If they are moving with power applied then I would think something is stripped if the wheels don't move with the gears.
I agree with you about buying old Tyco locomotives on eBay. You never know their provenance and they were priced new to be disposable. They can be rebuilt, but patience and experience with small electrical/mechanical equipment is vital.
That is not to say a novice can't do it. If you have good hands you can gain the experience through trial an error. I am not saying you do or do not have experience. I am speaking in general.
I have rebuilt several Tyco diesel locomotives but I had to buy 2 or 3 for everyone I was able to rebuild. In some cases I was able to find new parts available on the web.
The only reason to rebuild an old Tyco locomotive is for the love of doing it. It will not be cost effective in most cases. I have gotten lucky a few times and bought good running Tyco locomotives, but it's a shot in the dark.
I have rebuilt the ones I had as a child/young man. Lionel is easily worked on with parts readily available, but I have even had my fill of repair those locomotives.
Now I would only buy new unless I came across something intriguing to me.