Basically, it turned out that my old AirPort base station *did* have the "capacitor of death" problem, but it was actually the "capacitors of death" (meaning both capacitors blew). I hadn't looked carefully enough. Fortunately, my good friend Joel Snyder at Opus One had given me two replacement capacitors when he heard I had an AirPort, so I didn't have to go out looking for them when I replaced them. (Although, I have misplaced my soldering iron, so I had to go visit a friend for that!) The instructions on how to replace them were pretty clear. It's not for the faint of heart, but it sure beats paying $300 for a new AirPort.
The WAP11 goes back for a refund, and the AirPort base station is working fine again.