https://finalfantasy.fandom.com/wiki/Chemist_(job)
Technically you are correct based on similar terminology. But game play wise, the battle job, and crafter class are kind of different. Instead of Alchemist/Chemist, how about we call it Salve-Maker? The job has been around for as long as Samurai has been in the FF Series, it might not be nearly as popular in comparison, but it does definitely have a basis/job mechanic/ Fits the Healer Role. So it would not be very far fetched, if they brought it back. As far back as ten final fantasies ago, you can just call it a classic Job at this point, really.
Ranger/Hunter/Archer just used bow and arrow originally. Bard has pretty much absorbed that class
Although if i remember correctly, in FFV it did have an "animal friends" ability. but most people leveled it for X-Attack.(4x Attack)