What Lore? I don't recall any FF in the main number series (ie: not tatics) outside of 11 actually setting up lore for Blu outside of it being a job that learns monsters abilities and can be filled in as any role.
Why would they give BLU long cast times? If you're going by FF11 standards a good portion of their physical spells are actually less than a second. A good deal of their defensive spells are less than a second. There are also traits in the game obtained either by leveling or on gear, both in FF11 and 14 that prevent spellcasting interruptions.
No one cares what type of armor they use. Rune Fencer in FF11 (the same FF you base BLU off of) uses light weight armor as well, it actually shares it's non Job specific gear with BLU and it's one of the go - to tanks, It also has to rotate Phalanx, Crusade, Regens, and Temper up all of which take at least 3 seconds to cast. The same can be said about ninja as it has been a tanking power house since day one in 11 who had to use long casting times to maintain hate and defend itself, now it's a DPS in 14. Different games have different roles for each of the jobs. Cecil's DRK in FF4 being a tank? You mean the tank that kills itself to do AOE damage? Yeah no. If anything SE took the Tank design of Rune Fencer from FF11 and slapped Dark Knight's Aesthetics all over it. A tank whoms primary weapon is a 2-handed sword and who focuses on magical defense.
https://www.bg-wiki.com/bg/Rune_Fencer
Mages make for terrible tanks because they've all been designed this way currently in 14, this is reality. Just like how all jobs that were not design with the ability to main tank in 14 will die in a single tank buster in just about any raid. This is reality and your logic is flawed. Why? Because they could always design and implement a mage that can tank.
Sounds like you only played BLU during the abyessa era. As no one really did skill chains at the time of Blue mages release and nothing "proct" for at least 3-5 years later when Abyessa was released. During it's initial release the good thing that stood out about Blu was it's ability to chain head butt back to back and keep a mob stunned as well as being able to easily do a small burst of damage to enemies with extremely low defense that also drained their MP dry. This is why you'll never be able to find any end game content that took BLU to anything outside of instance events for that period (Limbus, Dynamis, Assaults, Salvage), where it could at this point in time Tank, Back Up heal, Mediocre DPS. In anything else they were forced to SA(force crit) cannonballs, which other well-geared and knowledgeable players destroyed them. However as they've given it more spells it's ability to tank, dps, and heal have also gone up with it eventually making it able to replace any role in the party assuming the BLU was geared for said role and had spells for said role.
Then why can it both heal and tank along side the the other roles you just mentioned in almost every incarnation of the job? Tanking mainly being relevant for FF11 since there was no way to really "tank" other than the flimsy and random cover skill in any other FF? Rather it be a tank or not is up in the air. But your reasoning for it not being able to be such is flimsy at best.
And this is the only valid point as to why BLU wouldn't be a tank in 14, and is more based around what lengths they'd be willing to go to actually make it into a tank. Rather than "if they can"