Well corrosion could still be the culprit but I think we should make sure the battery circuit is operating as it should. Personally I dislike desoldering parts from old PCBs like these, they can be very fragile. Trying to remove a chip can easily rip traces or pull vias, especially those already damaged by the spill. So that's why I ask to probe things first.
Anyway, that connection from pin 2 of the RTC/CMOS to pin 12 on 4069 (the pin count for 8-14 should be in opposite direction, they go counter-clockwise) is the clock signal. Datasheet for the HD146818 confirms it. Now, without oscilloscope you can't really see the signal, if it's correct or not, but even with volt meter you should be able to test it, more or less. When the PCB is powered there should be about 2.5V there (the actual value will depend on how the meter sees it because it's AC, not DC). If you see close to 0V or close to 5V then the clock is not running. Most likely because the 4069 did not start oscillating due to incorrect battery voltage. 3V battery might be too low for this board.
Keep in mind that the clock output will be lower when on battery. In general you should see about half the voltage that's supplied to the chip.
You can pretty safely connect anything from 3V (which you already have) up to 5V as the battery. 3xAA(A) pack will be 4.5V (3.6 if those are NiMH cells). An external DC PSU is OK too, anything will do really, the power draw is very small (way less than 1mA). So pretty much the voltage you see at the output when nothing is connected to that PSU is what it will feed to the motherboard.
So for this test:
- see if the RTC clock is running or not
- see if the 8042 keyboard controller is connected to the 4096 as well (It might be a different pin than the RTC though)
- see if both ends of R13 are connected to 4096
This will let us know where to look next.