Follow-up: I didn't fix my IIcx, and now think the problem is not the logic board.
I recapped my IIcx, then replaced UK2, UL2, UM2, D5 - D7, Q2, Q3, as well as the RAM bus multiplexors H1/2/3 - J1/2/3. These chips all showed signs of corrosion on their leads and I was able to inspect the pads...