We'd already power-cycled (and it's the first thing the automated system asked me to do), but disconnecting and reconnecting the cable while it was powered off was counter-intuitive to me. Why should that matter? I'm guessing that "off" isn't really "off" and there's something in the router that's still paying attention to the input, and only interrupting it physically gets it to reset whatever it needed to reset, but I'm reasoning from observed behavior; I don't know anything about router hardware.

People who will try to fix it on their own first will think of power-cycling; they won't think of disconnecting the cable. So it'd be great if Verizon would drop that hint somewhere in its support doc that Google took me to.