I'll concede on BLU, it was a dumb jab at what I feel is a tiny, if not irrelevant (and honestly, dumb) issue with Fishing. Fishing has got issues, having to buy bait is not one of them.
But to your point, no combat classes have to go out to a shop to replenish stuff. But it's present in other parts of the game. Every job has to replenish lost skills in Bozja. Using your argument, I would say that picking the right skills is a core mechanic of doing Bozja/Zadnor/DR/Eureka/etc. Should we also just make it so that you can just pick a skill and get rid of the amount of them you have? The unlocking? To get the skills I have to go and do the content in the zone, or I can buy the artifacts and hope to get it. They cost more to get than fishing bait, both in gil cost (if buying) and time. -- if that sounds really dumb, that's how the argument to remove bait sounds to me (someone who's fished up easily 45k fish).
Fishing bait is cheap. Need to make space? Chuck it. Not sure what to use? Ask someone, read the descriptions, check online. Unless your going for big/legendary fish, you only need 1 type of bait, the Versatile Lure. That's it. Pick up 20 in Limsa down by the Ocean Fishing dock and that will be all you'll ever need. Costs you 6,000 gil. I'll spot you. Heck, it'll even catch some big fish.
I'm just having trouble seeing the arguments against bait buying or why would people want to get rid of the need to buy bait - or even the need for bait in general? Is it Convenience? Gil? Too many types of bait to find? You just don't like Fishing and want it to be like every other DoL in the game? Do you want bait to be gatherable or earned through turn ins? There's an answer to all of this.
As Canadane says, Fishing is different from all other jobs, it always has been and always should be.
Post Edit:
I would also say that for some later game fish/patch fish -- earning the baits is a core mechanic too. It would nerf every legendary fish, and a good portion of big fish if we just were given unlimited supplies of the bait.