So far so good. The process is not very illuminating. They put on my estimate (which was emailed to me two days after the watch made it to the service center):
Customer requests
Double Jump, Hour Hand Loses An Hour Overnight, Date Wheel Issue
Necessary works
Check Movement. Invoice Price $0.00
So not sure what "check movement" will entail, if they'll replace it entirely, or end up telling me what exactly what was done, or do a general service, or what. But the good news is they appear to be doing it for free! On the online form to confirm the service, the note said "Customer out of warranty, service compliments of Rolex"
So... can't complain about that! They estimated 5 weeks. Now to play the waiting game. Will reserve final judgement until I get it back safe in hand, but so far it seems like they are taking it seriously.