Disk First Aid, especially of that vintage, doesn't check very thoroughly. I doubt that it would catch the sort of error that would cause the flag-clearing problem. You'll need a more industrial-strength utility that does reads and writes, multiple times.

As for the floppy theory, highly doubtful. Since you already know it's bad, it is a total non-surprise that DFA thinks it's bad, too. No data about shutdown status gets stored within a floppy drive, so that's not the first (or even second) place to look.