The OP mentioned collecting mobs; that is basically how it works in other mmo. In XI, if your pet dies, you gotta buy another or tame another. In other mmo, you simply select which pet you want to call. In WoW, you can pretty much tame anything but they shrink down to size when summoned by the player.
XI had some of the worst pet mechanics ever. Hopefully XIV follows what the OP suggested. You know how everyone walks around with a grasshopper in XI? We have to rely on them to release pets which are abundant in the wilds. We don't even have most of the families, because the pets are capped. BST is a mess in XI.
So it makes more to sense to I don't know.......Tame whats roaming the wilds instead of waiting for developers to release stuff already walking around in the game?
If it is a limited job, what is off limits? Are BST taming Spoken like the BST mobs from XI did the players? Skys the limit as a limited job. Group job, expect to see everyone walking around with crabs.
(Pun intended I guess....)