Should I have set my player's timezone to GMT before trying that patch? I would have figured the patch was smart enough to convert my timezone to GMT before sending the time.

The player should have a timezone setting, which has nothing to do with the actual real time clock setting in the hardware, by the way. The patch has no business doing time zone correction, because it's a unix machine: you set the time in UTC, and the timezone setting in software shows you the correct time for the zone you have set. When you change timezones, you change your view of the time, not the actual hardware clock. That's how my player works, and I didn't do anything special.

Windows is weird and encourages you to keep your hardware clock in your local time. I have so much hatred for them.