Most PC-style drives can't read the second side of "flippy" disks
That's baloney. The whole idea of a 'flippy disk' was to be able to read the second side of a single-sided disk. To make a flippy-disk you merely had to make an index hole in the correct position of the floppy's envelope and a write-enable notch in the correction position of the envelope also.
I made a cardboard template to mark those positions and used a hole-punch of this kind:
to make the holes themselves. Note that for the index hole, you had to carefully punch a hole in the envelope only, not the actual magnetic media itself.
Making a flippy disk was to use the back side of a floppy when you only had a single-sided drive (like I did back in the day).
If you put a flippy disk into a double-sided drive which is expecting to read double-sided disks, you're going to have a bad day. The data on the back side of a flippy disk is written backwards compared to how it would be written on the back side by a double-sided floppy drive.
If you already have a double-sided floppy drive, then you don't have any reason to make flippy-floppies to use that second side of the floppy.
To read the second side of flippy-floppies that you already have, you use the single-sided software that wrote the disk and that only uses a single side of the floppy and flip the floppy over. Then you transfer the data to a normal double-sided floppy.
That's baloney. The whole idea of a 'flippy disk' was to be able to read the second side of a single-sided disk. To make a flippy-disk you merely had to make an index hole in the correct position of the floppy's envelope and a write-enable notch in the correction position of the envelope also.
I made a cardboard template to mark those positions and used a hole-punch of this kind:
https://en.wikipedia.org/wiki/Leather_punch
to make the holes themselves. Note that for the index hole, you had to carefully punch a hole in the envelope only, not the actual magnetic media itself.