Exactly where did this this idea that Blue Mages are synonymous with swords come from? Where most of these concepts originate is Final Fantasy Tactics. A single game. Granted, it's one of the few that has a class system, but I remind you that in that game also Red Mages used swords. Rapiers, to be precise.
No, the idea Blue Mages used swords came from FFV when they could equip them. However, in that game, Red Mage was the better melee fighter as it had access to heavier armors as well as higher strength while Blue Mage had higher magic stats and a wide selection of monster skills to use.

Regardless, Red Mage is the original fighter-caster in FF and those that want to push it as mainly a healer tend to forget that white magic has always only been 1/3 of what it has ever done.