What was considered the best practice, or "military-grade" procedure, for protecting secrets stored on floppy disks when disposing of the old disks?
This document from the NSA, NSA/CSS Storage Sanitization Manual: Purpose and Scope, even though from 2014, is a good summary of how governments have been doing secure destruction of media for a long time: note that all the methods are very old-school! Other government agencies, both in the US and internationally, likely use very similar methods. As far as I can tell, the major changes have been in adding new types of media in the recommendations (and altering the list of manufacturers and devices.)
Regarding diskettes in particular, the procedures are detailed on page 5:
[3. Magnetic Disks: Magnetic disks include hard disk drives and diskettes.] [...]
1) Sanitization: Sanitize diskettes by using one of the following procedures. Remove all labels or markings that indicate previous use or classification.
a) Degaussing: Degauss the diskettes in an NSA/CSS evaluated degausser; see Reference b.
b) Disintegration: Disintegrate diskettes using an NSA/CSS evaluated disintegrator; see Reference d.
c) Incineration: Material must be reduced to ash.
d) Shredding: Shred diskettes using an NSA/CSS evaluated crosscut shredder; see Reference e. Remove diskette cover and metal hub prior to shredding.
So you can take your pick of degaussing the disk, shredding it in a paper shredder, disintegrating it (apparently thinner but longer shreds), or burning it. If you're really interested in what devices the NSA approves, they are listed here.