Fisher has "unique content" in the form of ocean and spear fishing yet we don't call it a limited gatherer. If you think anyone would seriously let a blue mage into a grinding party for old content, I have to question whether you've actually used party finder to unsync old stuff.

If a blue mage tried to meme their way into a Suzaku EX farm, I'd kick and blacklist them on the spot. Limited jobs have a negative name attached to them because they are objectively handicapped and cannot access relevant content in the current expansion, or even content they'd be good at in previous ones. Blue mage can graduate from being called a limited job when it can play with other jobs normally. Masked carnival is a boring slog to get through at best, all for the sake of a raise spell. Other than that, there isn't much going for it.

As for beast master, please if you want to add it just clone WoW's beast mastery hunter instead of making another system that won't survive past 2 weeks. 80/20 rule here. A minority of blue mages who insist on gatekeeping the job and forcing it to be "so special and unique!" that is turning off most of the playerbase should not be the sole voice heard on this matter. More people want to see full jobs than content with as short a half life as blue mage patches.