No, they've actually never been the tankier of the mages. Not in FFI, not in FFIII, not in FFV, and not in FFXI. They have on occasion been pretty good offensively, but they have never been good defensively. The best argument one could make is that the combination of all their buffs allowed them to basically live forever in FFXI, but that was less "tanking" than winning a war of attrition due to having endless MP.
White mage has been tankier than red mage on several occasions, though in most cases they both share the worst armor in the game. (This is probably due to the influence of D&D, where clerics wore armor and any class that had magic user spells didn't, hybrid or not. I've always assumed the original red mage is vaguely inspired by a D&D bard, also a light armor class.) This is actually most true in FFXI, which is where red mage is the most tank-like. White mage gets decent shields and even some honest to god heavy armor in that game.
So could they make red mage a tank? Sure. I mean ninja ended up being a tank in FFXI. Would it have any basis in an existing Final Fantasy red mage's role? No.
So your argument boils down to "it can't be a DPS, and it can't be a healer, so it must be a tank"? I mean that's kind of insulting to the concept of tanking. Should dark knight have therefore been a healer, since traditionally they had heavy armor and good offense, and thus couldn't be a tank or a DPS?
DPS are defined by offense, healers are defined by healing, and tanks are defined by defense. Red mages traditionally have had offense and healing, but never (despite the repeated baseless claims in this thread) defense. They don't wear heavy armor, they don't have a lot of hit points, and they don't generally have unique defensive abilities beyond those of the white and black mages they represent. I mean, I guess they have Phalanx in FFXI, so that's something.
I agree with you that red mage doesn't slot nicely into DPS or healer, but that doesn't make it a tank. I'm actually very curious (and a little concerned) to see what they do with red mage.