There's an issue on the FluxEngine git repo that may be related to this: https://github.com/davidgiven/fluxengine/issues/75
At the end of the thread, David Given reports that some PC floppy drives drop pulses when trying to read 800k Mac disks: https://github.com/davidgiven/fluxengine/issues/75#issuecomm...
So the Cambridge page might be saying "use an Applesauce controller" as a sort of shorthand for "use a drive from an actual Apple computer".
I received a response stating something similar on the git repo. It is still incorrect to blame the controller, and I suggested instead to provide some compatible drive models to use instead like they already do in the more generic sections of the article. It should even be possible to to use the Mac drives with the Greaseweazle with an adapter because the 800k drives from the Apple computers are just Sony drives with an optional eject motor installed.
Unlike the Apple II they do not have a custom Apple PCB. It's possible Apple requested some tweaks when ordering from Sony but I don't think they are all that special. So it should be entirely possible to read the disks with other drives. Based on the information in the page you linked stating any 720k drive is compatible, I suspect the higher read rate may be a vestigial feature that Sony over spec'ed during the introduction and standardization of the 3.5in disk.