Just when I'd lost all hope on the IIcx motherboard today, I decided to remove the RAM - one bank of which were like half-height 30-pin SIMMs that I was nervous about removing, trying not to bust the retainer tabs like what happened on the other bank.
Soft power on both keyboard and the rear button were still completely useless, of course, so I bridged pins 9 and 10 again. Yep, still the ol' Chimes of Death, but I would expect that due to the lack of RAM.
I put one bank back in (the bigger SIMMs that I think are just 1 MB each, possibly stock equipment), bridged the pins... wait, that's the normal Mac II chime! I'm getting somewhere with this thing!
Just for reassurance, I put the graphics card back on one of the NuBus slots (IIcx doesn't have integrated video), connected my monitor, jump-started it again... successful boot chime, working monitor output, floppy disk question mark icon because it couldn't find a System Folder with the drives disconnected (didn't want to reconnect those again outside of the case), this thing might just work after all.
I still need to figure out where the fault lies in the soft power switching circuit that makes it not respond to the proper power buttons at all, but I'm closer than ever to figuring out what's wrong with this thing and what actually isn't if it can boot fine when jump-started.
UPDATE: The physically larger SIMMs are indeed a mere 1 MB each, but the IIcx boots up fine with 4 MB of RAM. Floppy drive heads need some cleaning. The system doesn't seem to detect that the keyboard power button or rear mobo power button have been pressed at all, but going to the Shut Down menu option shows the "It is now safe to turn off your Macintosh." screen, as if they might've foreseen this sort of issue.
This gives me some loose ideas about what's wrong with the startup circuit - that the motherboard somehow isn't registering that the soft power buttons have been pressed in the first place. It's not the power switches themselves (the contacts switch accordingly), so I guess it's a matter of going through the schematic and seeing what is or isn't getting the connection it's supposed to be.
Reinstalling the second bank of RAM also results in a successful boot, but it's still only picking up 4 MB of RAM; it's like it can't see the Bank B slots for some reason. How peculiar.