Bizarre...I wonder why this happens to some people...be interesting to understand why things get messed up in the first place.
Easy. I was messing with my phone after the first day's horrible battery life to turn off "useless features" (turn down the light, turn off wifi [no local hotspot], turn off GPS, turn off whatever those other buttons are just for good measure). Then, at some point, I cleaned up my screens and removed that widget. Since there is no way to turn synching on without that widget on the screen, there is no way to stumble upon the solution.
It's even more horrible since, apparently, older Droids have a way to turn it on without the widget. That makes it a problem that only first time owners of new Droids get.
Verizon should be mortified that one of their workers couldn't fix the problem on the prior poster's phone.