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, and at that time?
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.
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 obsolete short after. Kind of if Porsche Volkswagen would introduce a whole new gas engine design today. 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.
As history goes, the 80 column card survived a bit longer due its massive instalment bade.