Hans sent me a factory EEPROM file, and it loaded onto the QDX in bootloader mode, but unfortunately the serial port is still non-functional.
Is it documented anywhere what the 5 header pads are at the top left of the board?
I was checking through the board with a voltmeter for good voltages, and decided to check what was on those 5 pads. To my surprise, when I touched pad 3 (circled in the image), the QDX switched into bootloader mode! This was with a digital multimeter, so it was presenting a very high impedance (to ground) at its probe. I unplugged the probe from the meter and tried it again (so I was touching it with a probe and a dangling lead, attached to nothing), and it switched back into normal mode. This is repeatable, and touching that point with a disconnected lead will toggle the QDX back and forth, in and out of bootloader mode. Really unexpected.
My guess is that the brief capacitive loading of the lead was enough to make the voltage dip momentarily, and so the microprocessor switches modes because that dip is enough to trigger it as a power cycle. (I repeated this with both a big bench power supply, and a portable battery pack.) And it makes me wonder if there's an issue with power stability on the board.
Leigh KG7WED