I was able to flash the drive with FlashFloppy firmware with no serious issues.  There was a slight problem of figuring out when to connect/disconnect the reset line, but a little trial and error got it working.  The instructions provided were not quite right for me.  I had a couple issues:

It didn't get detected as a drive at first.  The instructions had me put the jumper on the wrong pins.  I moved the jumper down one set of pins, and the drive was now detected.  However, opening the contents looked like garbage folders and filenames.

I wasn't sure the pinout of the cable was correct.  Some people have the drive select pins reversed on their cables. I finally cut traces, and soldered on jumper wires to swap the pins, and got the same behavior.  I should investigate this more, but I suspect the same signal may come out on both pins.

I then loaded the logging firmware to get more insight.  The log said it was only loading the autoboot drive image and not finding any other images.  I switched from the HxC compatibility mode to the indexed mode where I can manually specify the disk images, and bam it worked.

I think the HxC compatibility mode must set up its own sorted list of disk images, but it only works when you boot into it.  You can't run that program by opening the disk and launching it.  I think that's peculiar that some disk directories can't be read, but they can be booted.  Seems inefficient and awkward.  I can see some software which requires a cold boot, but you can always keep a directory structure, and give an error message if the software was loaded manually.  So I won't be able to use the HxC mode until I can boot from this drive.

Anyway, three of the four boards are now working.  I may either look into modelling a 3D printable enclosure for the drive to make it neater.  Or I may look into the more dangerous task of disassembling the computer and desoldering that chip for the drive swap.