Soft-power MOSFET circuit leaks - what to do?
Sure enough, it was leakage from the power MOSFET pull-up resistor through the controller's pins, though those pins' protection diodes and then to the rest of Vcc.

The new circuit uses a common cathode diode array for each button and an N MOSFET on the controller's pin all of which isolate the !PWR net from any potential leakage paths.