Boy, I'm good at messing up transistors!  This time it was just the pin
numbers for the collector and emitter reversed, because I didn't copy
the symbol from the datasheet but rather assumed another SMT 2222 would
have the same pinout.  Interestingly, it actually was able to control
the MOSFET at VBUS = 5 V and at VBUS = 9 V, but at VBUS = 15 V the
reversed transistor couldn't pull the MOSFET gate down enough.

Tested to probably not blow anything up if assembled properly!