In addition to the internal floppy drive that came in my Apple III, and which does not work, I have three additional spare internal floppy drive mechanisms. Today I tried all of them, and found that none of them work well enough to boot an Apple III disk written from a disk image file by my Apple IIe. The disk format is identical to Apple Disk II 16-sector 5.25-inch format, and the Unidisk 5.25 drives I’m using on the Apple IIe haven’t had any trouble reading and writing any other disks from a large collection from various sources, so I’m pretty sure the drive on the Apple IIe is not the problem.
Having four non-working drives made me think that perhaps the floppy controller, an integrated part of the Apple III motherboard, might be bad. I switched to my spare motherboard, and didn’t have any different results.
I also have an external Disk III drive. I tried cabling that to the internal drive port, in place of the internal drive, and was able to successfully boot the Apple III Dealer Diagnostics disk. That mostly worked, and I was able to test the RAM and ROM of both motherboards. Both are the later 5V configuration, one with a 5V 256KB memory board, and one with a 5V 128KB memory board (which could easily be expanded to 256KB).
I say “mostly worked” because the drive did give I/O errors trying to load some of the diagnostics, or to display a CATALOG of the diagnostic disk. So really, I have five Apple /// floppy drives that aren’t working properly.