Well, there's generally two issues with this line of thought:
First would be that a chemist would be physical and "item reliant". With Item reliant I wouldn't been being bound to healing supplies, but rather than their healing comes in forms of items - Potion Pitching if you want to call it. Red Mage, on the other hand, is magic reliant, as the name implies, and to an extend physical based. The class concept of Red Mage and Chemist would clash and create a class that would be magic based as well as physics based (pharmacy).
Secondly would be the the weapon affinity of the classes according to the design patterns. Common weapons for chemists (or alchemists) in the Final Fantasy series would be a gun, mace or dagger. Red Mages, on the other hand would have affinity towards fencing swords, such as rapiers, sabers and degens. They also have some affinity towards daggers. While they have a common weapon affinity, the rogue class already claims the dagger weapon in Final Fantasy XIV. As long the armory system stands, no other class can use daggers unless Square-Enix introduces a seperate category for daggers, like how they did that with the initial daggers for Gladiators.

In other topics, the possibility that gunners will become chemists later has been discussed. I don't think this will happen personally, but that's for another discussion :P