Hey, since you guys were so kind to respond, I thought I'd give some followup. I remembered I had a dusty but functional ATV 2nd gen sitting in the basement, so I opened that, and took out the (known good) power supply and put it in the 3rd gen. (I reasoned this would work since a replacement power supply from China lists it for the two models (three actually) and it looks the same.)
Anyway, didn't work in the gen 3. So it may be the main board. But...just for kicks, I put the Gen 3 power supply into the Gen 2 to see if the PS was good, but that didn't work either. Still dead.
So I think maybe _both_ the power supply _and_ the main board on the Gen 3 are kaput. Fortunately, when I reassembled the Gen 2 with its original power supply, it worked.
A lot of futzing for something I see you can buy off eBay for ~$15, but the puzzle was interesting to play with, and more experience always helps. We don't need too many bells and whistles at the lake place, so I'll just move the 2nd gen out there.
Thanks!