So it turns out that there is a bug with Garmin Connect Mobile 3.9 that turns off audio and vibration alerts on my 735xt after a sync. In other words, if I turn those two settings on (Menu -> Settings -> System -> Sounds -> Alert Tones On/Off and Vibration On/Off), they are turned off after syncing with my iPhone running GCM 3.9.0.8.
Brilliant.
I'm seeing people reporting issues across several device forums about sync issues, and some believe it's related to the latest firmware for their devices, but I'm convinced that it's specifically an issue with the latest GCM.
I called Garmin, and they don't seem inclined to believe that there is an issue with GCM 3.9, so the more people that call to complain about this, the better.