In better news, the keyboard cable showed up, and the keyboard appears to be working fine when the machine is powered up (I can navigate the boot menu and such). I still have to heat up that area of the board before I can power the machine on, though. At least now I know the specific area I have to heat up and I can try replacing that comparator first when my new parts get here. I'd try putting my one extra back in again, but honestly, it's a DIP component, and the ones on this board are all surface-mount, so I basically had to solder up 14 individual wires to the thing, which couldn't have been helping the situation any.
EDIT: Yup, I just replaced three more capacitors, this time on the main PCB, near the chip I've been heating up, and I still can't power up the machine unless I directly heat up that first comparator.
EDIT 2: I'm giving it one more shot with swapping the comparators on the board. I had already swapped two, now I've swapped the third one for one of the others, so none of the comparators are in their original locations now. It's powered up, but I don't know if it's the swapping that helped, or if it's once again just the fact that the comparators got heated through my soldering iron.
EDIT 3: Yeah, either multiple comparators are bad, or it's something to do with the board at around the position of the first comparator. Even with the comparators swapped out, I have to heat the board in that position for the computer to power up. Another interesting note, what I've done to the board so far seems to have enabled it to stay powered up for longer periods of time, although if you turn it off it still has trouble turning back on until you heat the board again.