What was the use case for the 96-column punch cards introduced with the IBM System/3?
About the same as for introduction of 5.25 or later 3.5 inch floppies: same capacity in a smaller form factor. In fact, the 96 column card was kind of the 3.5 of punch cards, just right to be put in a shirt pocket.
So ... why did IBM invent them,
To give more storage in smaller space to improve handling of small to medium data sets. A 96 column card was only about 1/3rd the size of a standard card but could hold 20% more information, so that's almost quadrupling data density.
Of course, when just looking at the increased storage capacity, one could as well have used the existing format and tripple data density. But to use this software would have to know that cards now three times as large. But 80 columns was deeply engraved into the very foundation o all databases at the time. Changing all of that would have been an incredible task.
Making the card smaller on the other hand would bring the same improvement while being mostly compatible on the software side. A 96 column card can quite well be used by software expecting (and producing) 80 column cards. It would simply not see the additional 16 columns while on output they stay empty.
No changes handling required but an immediate save of 66% storage space.
and at that time?
Time is maybe the most important item here. The System 3 was developed in the late 1960s - introduced 1969 - right at a time when punch cards were still a thing but started to become more and more obsolete due tapes and disks.
It was almost as if Volkswagen would today introduce a whole new (*1) gas engine design. It would be awesome tech and great engineering, but vanish faster than its predecessors as we're moving toward electric propulsion.
In fact, essentially the same happened with the 96 column card. it got introduced in 1969 with the IBM 5424 Multifunction Card Unit (MFCU) and replaced starting in 1973 by the IBM 3741 floppy units. First as add-ons to existing machines, later the MFCU was no longer offered, so 3740 floppies became the way to handle small data sets (*2).
As history goes, the 80 column card survived a bit longer due its massive instalment base.
*1 - As in 'real new super high tech 30% more efficient'
*2 - Noteworthy again that the floppy did only play a small role here as it eliminated the use case for small data sets. Even before floppies disks and tapes did successfully start to replace punch cards for large data sets. so to floppy just killed the last niche for punch cards, and even an improved one couldn't hold much against that.