Status update:
First off, wow Patek in New York and Richard (thanks) was super helpful, very thorough and patient.
I feel much better discussing the process and gaining an understanding of what the service will involve.
Patek has my watch and will issue a notice of receipt. Then they photograph the watch, note the serial numbers and perform an inspection. I will get a letter for each task.
The entire process will take 21 weeks
at this point for a full service, disassembly, cleaning, repair of problem with the winding stem, re-assembly, quality check and regulation.
The quoted cost at this time is $900 for the service (dependent on problem and parts needed), $380 if I want a polish (they will inspect and advise) and $50 for two-day Fed Ex delivery.
I will continue to update this process.
Thanks.