Lots of scrubbing and cleaning later, especially of my bathtub.

The IIsi wasn't as dirty, but leaking SMD capacitors started to do their evil thing.

First a general recap of all electrolytic capacitors was on order for both of them, and while I was at it, I also replaced the tantalums.

I also replaced a couple affected resistors and transistors.

The power supplies had a couple obviously bad capacitors. I didn't bother shotgun replacing them. The owner wanted to outright replace the PSUs with modern ones.

The Macintosh II line however puts +3..6.5V from the internal batteries onto the power supply's /PFW line. Once the power supply started up, the circuitry keeps this line high with power from the supply.
Seems a bit convoluted, but it boils down to having to invert the /PFW line using the standby power provided by the ATX supply.