The little Shuttle boxes are packed quite tightly so it was not entirely straightforward to get the power supply extricated. But when I did, I noticed that it made a little sparking/clicking noise when the power was applied. Also, all the +12V outputs seemed to be between +6V and +8V. So, diagnosis confirmed. Fortunately I was able to just look up the power supply model number on Google and find many people willing to sell me a replacement.
I was also able to confirm that falling voltage is a common symptom of failing power supplies. But, I'm not clear from an electronics perspective why that should be. I've put together a switching power supply or two from components, but just DC-to-DC. My gut feel is that the order the components would fail in is: (1) fan, (2) "switch", (3) rectifier, (4) capacitor, (5) inductor. But I don't quite see how a switch failure would still provide DC current, just not enough. Maybe it's the capacitors that tend to go first, and thus they don't hold enough charge to reach the desired voltage. I wonder if anybody has done a study... maybe I should ask Patrick, our platform guru. Or my father, an electrical engineer.
ETA: I found this rather dense and jargonish PDF presentation from IBM. Their failure percentages are: ~30-35% MOSFET (amplifier?), 10-15% Choke (inductor), 10-15% Driver (switch?), 0-5% Ceramic (capacitor), 0-5% Poly (poly-propylene capactior), < 2% Fan. So I guess I was totally off base on fans and inductors.