It messes up party balance. DPS can be interchanged to a degree; a DRG in place of a MNK or NIN is not going to drastically change the party synergy because you have a class replacing another in a role. You're asking to add the equivalent of a fifth wheel, and one that would make putting together a party that much more of a pain.
Let's also not forget that "support" has a tendency (on top of being difficult to balance and being either useless or mandatory) of being less populated than tanks and heals, which means that 30 minute duty finder queue would become even worse if you had to fill 4 roles instead of 3. Even outside of DF you have the issue of still requiring said fourth role for raiding. And because of the established pattern of support being less populated, you have a very bad scenario in your hands. Look back to the past existence of princess bards and princess red mages in FFXI if you don't believe me.
So you're asking to be different for the sake of being different. Not because it's a needed dynamic (because it isn't needed), not for the sake of improving gameplay, but entirely based on a desire to do something different from the trinity. That's not a good reason to make any changes or inclusions.if they made it no better or worse than another DPS through their buffs/debuffs, it would be providing a equal level DPS class with entirely different party mechanics some players might find refreshing and enjoy.
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At this point I have to ask "why?" Yoshida himself has talked about sticking to the trinity. Reasons as to why that's a good call have already been covered. Reasons as to why wanting to shoehorn a fourth role for the sake of being different is a bad thing have also been covered, but some people seem to either ignore the arguments or simply don't care for balance and parity between jobs. It's really mind boggling.