I don't think Blue Mage is a "limited job" because they want to not give it attention; I'd argue that with the Masked Carnivale, the Blue Mage Log, and a ton of Blue Mage specific achievements, there's a solid chance that it's received more job-specific attention than any other job in the game (limited or otherwise). I think it's limited because they wanted to do Blue Mage with actual Blue Mage mechanics.
Blue Mage's whole core conceit is learning spells by seeing them cast by enemies you defeat. It would be extremely difficult to both have BLU able to learn things like Tsukuyomi's "Nightbloom" and still have them able to participate in real content. Not because the primal spells are necessarily OP compared to other job abilities, because of the versatility. Who'd want a black mage as the caster in their raid static if you could have a BLU who can use attacks they learned from the fights on previous raid tiers that might well trivialize specific mechanics, and act as a backup healer at the same time? (As the joke went with the Endwalker summaries of jobs, BLU sitting here going "Laugh all you want. This expansion I get Light Rampant.")
So, okay, you make a 'real job' version of BLU, and restrict them to the GCD spells that you learn, stripping out all the potentially-difficult-to-balance oGCD ones you learn from trials and raids; prune the list quite a bit, to make it so the spell sets are more or less balanced with the other jobs. Or something else to make the spellbook not difficult to balance alongside other jobs in content. At the very least, adding minimum levels for the spells, where you can't use a given spell when level synced down, so that BLU at level 50 doesn't have multiple hotbars of abilities where most jobs have only a few.
A bit disappointing (and potentially an immense pain if you take a hotbar into leveling roulette and end up at a level where you don't have any of those spells available), but it's at least still BLU where you go deliberately learn the spells.
But now you have the scenario where you get a BLU in roulette who hasn't learned some key spell, and the party's upset about it. (One need only look at the commentary when someone shows up in content without their job stone -- or having forgotten to do some job quest that a key ability was locked behind -- to see that the problem's likely to arise.)
Okay, so to avoid that, you just make sure that BLU gets the spells, guaranteed, at specific levels; no worries about someone having not unlocked that spell yet. But I question whether people who enjoy playing BLU would still like that BLU... because, to me, that doesn't seem like Blue Mage anymore; it just seems like another caster DPS.
Edit to add one additional note: if BLU becomes a full, real job, you likely end up with the same scenario as all the other real jobs... namely, you get maybe 2-3 new things per expansion. It would hardly be balanced easily if you still had 10 new abilities to learn from the various raids and trials over the course of the expansion, while another caster DPS gets an upgrade to one existing spell via a trait, one new oGCD, and then a capstone ability when they hit cap for that expansion.
BLU is a limited job not because it doesn't get enough love, BLU is a limited job because it pretty much needs to be in order to actually be BLU. Or, at least, to be a proper incarnation of blue magic.