I was under the impression that hairball remedy (whether it be a paste
the cat licks off your finger, or a crunchy treat that hides a gooey,
hairball remedy center) was supposed to help prevent the cat from
barfing up hairballs. I thought it was supposed to ease the passage of
cat hair down the feline intestinal tract, so kitty could poop out the
hair.

Am I mistaken in my understanding?

My experience has been that feeding my cat hairball remedy is more
likely to CAUSE him to barf up a hairball sometime within a day after
feeding it to him. Is this its true purpose? Much like the way that
eating grass can cause a cat to barf up a hairball?

Incidentally, I have a medium hair cat who gets combed nearly
everyday. He barfs up a hairball about once a month.