So I ended up writing a Basic program on the Atari to read data from the disk sector by sector and paint it on the screen (with the large 4-color pixels of graphics mode 3). The Atari was connected to the TV card of my PC and a Delphi program I wrote was running on the PC that kept taking screen shots and trying to decode the data from there. I quickly learned that empty sectors threw off my pixel position calibration so I added a mask pattern and a checksum. The sector address was also included. With that, I was able to transfer all my disk contents to my PC. To this day I consider it my greatest engineering achievement :)
Some ten years later, I went on to build an SIO2PC program called AspeQt. A more up-to-date community fork called RespeQt is still the most popular tool in that category used by the community. It even has its own subforum on AtariAge[1].
[1] https://forums.atariage.com/forum/184-respeqt-sio2pc-softwar...