A HD drive can read a DD floppy. The problem comes with writing. The HD head is half the width of the DD head, which means a HD drive just skips half the tracks when writing a DD floppy. Unfortunately, if the floppy isn't totally blank, this means the wider DD head sees both the narrow track written by a HD drive *plus* what was there before and not overwritten, and so can't read it properly.
I also used to find (in Atari ST days) that there were problems using HD floppies in a DD drive (even if formatted as DD), but I couldn't entirely understand why that was. Perhaps they used different magnetic materials, a bit like regular vs Metal type cassettes.
The issue with Double Density (DD) and High Density (HD) floppies isn't so much the head width, rather it's the nature of the magnetic surface of the disk, and its coercivity (strength of magnetic field required to write to it). If you mix and match systems, you can sometimes find that data you thought had been overwritten correctly can "revert" after a few days or weeks.
Head width relates to a different issue: in the days of 5 1/4" floppies (possibly the older 8" too), early drives could put down 48 tracks per inch of disk radius. Later drives with improved stepper motors and narrower heads doubled this to 98tpi. By double-stepping, a 98tpi drive could read and write a 48tpi floppy disk. But if it were to overwrite data laid down by a 48tpi drive, it would erase/write data in the middle of the wider 48tpi track. The new data would read fine in a 96tpi drive, but if taken back to an old 48tpi drive, the wider head could pick up a mixture of the wide, old track and the new, narrow track, giving read errors.
There were other issues with 5 1/4" too, including hard vs soft sectoring, encoding data with FM or MFM modulation, and other details that escape me at the moment.
You might be right about the media, but drives are still knocking about, used if not new. If the only purpose is to read disks sent in by others that'll do
I've just done a mental tally of the various floppy drives in our house: ten! (5x 5 1/4", 5x 3 1/2" of various denominantions)