Victory!
I was inspired to try fiddling with the Jasmine MegaDrive again on Saturday. To my amazement, I got the drive to show up in SCSI Probe, then a little later got the first disk cartridge to mount on the desktop. About a dozen disks were successfully copied with no I/O errors. They were last used in 1994, according to the modification dates. The Jasmine driver is stored on the disks themselves, so no additional software was necessary.
I think my problems earlier were from SCSI Voodoo, as I had to switch my terminator and SCSI cable on the back of the Jasmine box.
This has been a 10 year recovery project, as I originally got these MegaDisks (they are OEM of DTC TakeTen) back in March 2009, and only got the requisite drive in November 2015.