I would say that my first argument to Severian's basic argument that we only want Mystic Knight to retread old ground covered by other classes by suggesting that if we start by believing that it doesn't need a sword, that opens up a whole slew of design options. I usually go with a whipsword, or a double ended sword (You usually hear "Judge weapon" or "Zidane's weapon" here, but I think the most iconic explanation is Darth Maul's weapon, which should be kinetic and flashy enough to work as a weapon) but I can see the argument that these are just more swords with extra steps. I've also heard a mace, or axes, going with a sort of Viking esk rune magic sort of vibe, though I think those extra steps are important.
Second, beyond changing the weapon, you could in theory change the application of magic. I'd be cool with fire ice lightning, but if we assume these are overplayed, Spell blade from FFTA2 used status spells as it's main effect so there is precedent for non elemental magic, if they needed that wiggle room. (this is not saying that they should use those status spells for their blade magic, just that it allows for a break from the elements as a required feature). I don't think elements are a deal breaker so long as the weapon choice is creative enough, but not being elemental isn't a deal breaker for me.
Now, to address the "You've already got DRK/SAM/GNB if you want to be an elemental focussed melee combatant". Besides the fact that you chose sword classes because obviously all we want is a sword mage that goes whackity whackity whack with our Fire Ice Lightning weapons because it is us that lacks any imagination, I don't feel that these answer the desire for our job fantasy. There is more to what we want than just the colour of the particle effects that enshroud our weapon. We want to be a magical warrior enchanting our weapon with spells to gain power and effect (this is where I usually mechanically distinguish the Job by making the chosen magic imbued into the blade change the core combo, rather than just "more power"). Dark is a angry knight that taps into that anger to empower their weapon. Not a mage using spells to achieve the effect. Samurai is a martial artist swinging their sword, their battle prowess giving their swings visual flair due to... I'm not entirely sure to be honest. Red Mage is a long range caster with the odd DPS phase. Moving on. Gunbreaker is the worst example because it's not creating a fire blade. It's creating explosions by detonating enhanced shells (which at least based on my VIII understanding are actuallt just creating vibrations down the length of the blade and not damaging the enemy themselves, but whatever.) That'd be like me throwing out your Cannoneer example because BLM and RDM both already create big explosions. (also Scholar, Summoner, what have you.)
In short, they can change it up beyond just "dude makes their sword go whoosh fire and hits the enemy", in the same way your suggested water mage wouldn't just be a Black mage with blue particle effects instead of Red... and a different shade and shape of blue.