The price tag is certainly higher (~100€), but AFAICT you can use a KryoFlux to read everything (even strange low-level software protections):
Works with all major 3.5" and 5.25" drives
Works well with selected
3" (e.g. Amstrad FDI-1) drives.
Also works with 8" (e.g. Shugart 851;
might require additional adapter) drives; other types of ...
Instead of the board you suggested or a KryoFlux, maybe build a FluxEngine.
You just buy the specified FPGA development board, solder one row of pins from a floppy connector onto it (or, in a pinch, a row of pin header), flash it, and you're done.
You'll also need one of those wall-to-Molex power bricks that you see bundled with USB-PATA/SATA adapters, but ...
The firmware on all the 3.5" USB floppy drives of relatively modern make I've looked into seems to present the floppy as a LBA device with exactly 2880 sectors (of 512 bytes) to the host computer. That is the very same thing as an USB thumb drive (and in fact, most OSs seem to see such a thumb drive. You can even partition such a floppy).
Replacing the ...