I'd think this is likely to be a relatively simple problem, if I understand your description, with a mechanical latch in the winding mechanism, also having to do with double-exposure prevention. Sounds like it's just not latching for some reason, so either the shutter is not getting cocked or the system doesn't "detect" that it's cocked... behaving like the automatic advancing to the first frame just never detects that it's arrived...?
I have one of these cameras, bought used at a camera swap meet, and I did not discover several problems until later. But your film wind issue was not one of them. In my case, the RF was "sticky", the meter response was non-linear, and there were broken internal plastic parts in the lens mount. A professional repair shop in Seattle advised it was infeasible to repair, so I set it on the shelf.
I later heard online about Frank Marshman, an expert in Fuji repair. He accepted the job, fixed everything, and charged $135. Which I thought was a bargain, as it now works like new. You might give him a call...
Camera Wiz Camera Repair
169B Pleasant Hill Road
Harrisonburg VA 22801
800/471-8133; 540/434-8133