It was the router. Its replacement - same make and model - works as advertised.
It's quite a relief. That's all four family members without wifi for their various and sundry devices, for over 24 hours. Those with addiction to said things (like Younger Daughter - heh) had to go cold turkey. :)
That said, all is back to normal. And we, who aren't addicted at all (wink, wink) can go back to Netflix and watch Bones.
Having said all that, it's a bit disturbing that one can plop down $120 for a new device, and have it fail spectacularly right out of the box. It seems quality control isn't really a thing at Netgear.
Ah, well ... caveat emptor.