You know how you can puzzle over something for a while and the answer only comes when you ask a real person the question? I hit "post reply" and went for my lunchtime swim - no sooner did I get outside than the answer dawned on me.
I thought the diode across the capacitor would do it, since when power was on it acts like a non-conductor but when power is off, it would short the cap in the other direction. DUH! The cap will still have + on the Vcc side, so the diode will still be non-conductive - there is no "other direction".
Then I realized what you said. In a real world circuit, there'd be a sink for the current and the diode would flow in that direction, discharging the capacitor. I was confusing myself by thinking of the PoR circuit in isolation - if the power (and everything else) is removed instantly, the diode and resistor connect to nothing and there's no sink. That's not what happens 99.99999999% of the time.
Thanks for confirming!
I'll never get over how often asking a question can open one's mind up to a solution.