I'm sorry, but where does this idea come from that Beastmaster *MUST* have pets collectable around the world to essentially just be "FFXIV Pokemon: Gotta catch em all!" when Summoner exists as a pet-type job with a certain selection of pets you unlock / obtain canonically in their story line? Even BLU's ability learning couldve been questline gated, such as learning Bad Breath from a Morbol threat in their storyline or Magic Hammer learned from a quest-based "Apanda" (O3S add name, don't know the previous one), Martyn congratulating you on your victory and then you accidently slap him with a hammer as you shake his hand.
Beastmaster does not and should not require to become a Limited Job to justify its exstence as a pet-type job. It is in my opinion bad enough that one full job people looked forward to play in regular content (you can't even use it for MSQ quest instances) has been delegated to side activity status when said side activity could've been encouraged via normal party compositions. What we have now with BLU is likely never going to change, but let us not kid ourselves that BLU had to be this way for "tradition", because by same tradition Black Mage isn't abusing elemental weaknesses, Bard is hardly a bard and rather a singing bowslinger, Summoner for the longest time was not an intense summoning job like other FF games and Red Mage isn't nearly the "jack of all trades, master of none" as other games, something BLU could've done the same.
Additionally, blue magic learning differed from one FF game to another, FFXIV could've incorporated ability learning (and balancing) in a more structured manner or at least have three different types of BLU soul crystals / kits befitting a more designated role for BLU so you could still have BLU an actual jack of all trades, just not all of it at once.
My point is, Beastmaster could be a melee or ranged job with the usage of a pet similar to Summoner. I mean even Dark Knight and Machinist technically have pets, even if its just one right now, yet Machinist had two different auto turrets once, so there is that. You don't need an infinite amount of pets, three to six could be fine enough to create intricate gameplay.