Apple File Exchange should run on any Macintosh computer with a built-in 800K
disk drive.
The problem here is not AFE, but the disk, so the manual's statements about
the File Exchange Utility are beside the point. It is not possible to read an
IBM-formatted (MFM) 720K disk in a Macintosh 800K drive. Only the Apple
SuperDrive (formerly Apple FDHD) has this ability.
The Macintosh SE you used for testing is able to read a 720K MS-DOS disk, it
must have an Apple SuperDrive installed.