...What? Are you really trying to equate a fantasy MMORPG to real life to form an argument? And secondly, doctors don't make or augment medicine, they prescribe it based on a diagnosis.
Because you are equating that alchemist and battle chemist are the same thing when in reality they aren't so I am presenting a real life example of how to think of it. Sure doctors prescribe the medication but if you were to take whatever you thought was good enough you would probably hurt yourself or kill yourself. ALAS why chemist could be a medicus because they have knowledge on how to take more advance medication and how to combine it with other medicine for glorious effects.

I'm not even arguing the name
The name was brought up because many people have discuss how alchemist = chemist which isn't true, they are similar ill give them that.

more how the mechanics of the job don't really work with XIV's gameplay. Chemists fool around with stuff in your inventory to make either buffing or damaging effects. Works great in a turn based RPG where you can open up menus easily. Strip that away, and you're just going more for the aesthetic than the actual function of the job.
And with that logic why don't machinist and bards have to buy bolts and bullets? I don't see anyone complaining about that ruining the aesthetic and function of the job. What people mostly want with a chemist is a job that aesthetic uses potions and has the mix command which can be accomplished with some type of system like the mudra system.

Doctors don't make the medicine itself any more effective. Someone else pouring a potion down your throat doesn't make the potion any more significant than if you drank it yourself. Lobbing it at someone would affect its potency, admittedly, but only for the worse.
But they administrate the medicine and doctors can prescribe two or more medications to make them more effective which is essentially what the mix command does. If we didn't have doctors/medics we would have alot of more dead people around the world due to overdoses and such.