I sort of lumped utility and support into one group, but what you are suggesting is still inline with what I could see Red Mages fit into, being that other "little something extra" class.
Again the problem is what would they be classified as in the game? Tank because they can off-tank really well? DPS because they can DPS too? Support and Utility aren't currently a classification in the game and as I stated earlier if these were their own classification in the game I would completely support Red Mage being those, but since they aren't I tried to fit Red Mage into a classification that exists and could use a little more variety.
The whole support/utility class worked better in FFXI but FFXIV is very "Holy Trinity" role based and it seems that for a class to work in the game it has to be geared more towards one of those roles.
So to expound upon my Red Mage as a tank idea and show its more utility bent, here is what I was thinking.
Weapon would be a Rapier/Dueling Sword style of weapon with a Shield in the off-hand for added blocking defense.
They would essentially be able to swap roles between Main Tank and Off-Tank/DPS.
Main-Tank would rely on Magic Wards and buffs that the Red Mage would place on themselves that would help absorb and mitigate damage of different types such as (physical, elemental, etc.) and they would switch between the different wards depending on what the situation demands.
Off-Tank/DPS would rely on Debuffing enemies so that they take more damage from the parties attacks as well as buffs that increase the Red Mages damage output.
The Red Mage would only be able to be in one of these modes at a time, similar to PLD switching between Sword and Shield Oath.
All of their abilities should rely on MP more than TP to give them more of a mage feel.
Also, as I brought up earlier, it might be interesting if they had an ability that switched around MP and HP as a way to add another dimension of resource management during a fight.