So I figured out that the reset button circuit (R5 and the 2-pin header in the previous posts' schematic) were causing RST to short ground. So, I figured I'd build out another board and see if not populating those parts would get me anywhere. So I did, plugged it in and hit the power switch and... nothing. Not even any lights on the digital I/O board.
So I checked the voltages and it turns out the batteries had dropped just a tad too low for the L7805. I charged them up, made dinner, did some other stuff, and then tried again.
Still nothing, not even LEDs.
Meter was reading abnormally low voltages, and I touched the L7805 and wow, that sucker was H O T. So I've got a short somewhere. I figured it was just a defect in the board, and sort of resigned myself to really needing a new board. While I was here, I figured I could pull the parts from my parts bin for a new board, even though I'd still have to wait. Then I realised something.
One of the 74HCT545's was backwards. That'd sure cause some problems, wouldn't it?
So I fixed that and power on and all eight LEDs on the digital I/O board are now lit, but I still don't get anything on the serial. So there's that to debug, but I think this is in a place where I might actually be able to continue making progress. I still don't like how thin the PCBs and I'd like more room to make it easier to replace the EEPROM, but this works for now (I think).