You probably -could- play Arcanist without using Carby... But the real question is why you would want to do that.
Compared to Conjurer and Thaumaturge with their Potency: 140~150 magic, an Arcanist's Ruin with a potency of only 80 is really going hurt you if you dont use carby for the extra damage. You get some nice debuffs, and you could technically be a healer. But you could be all that with Carby out as well.
For people like you (and well, me) who dont really like to use Carbuncle, if you set him to "Sic", he will attack what you attack automatically and use Abilities as soon as they are ready. It removes the tactics people can implement that actually hotbar all his attacks, and in general you will do a little worse, but you can pretty much ignore Carby for the entirety of your Arcanist life up to 15 or so. He'll do whatever, and you'll benefit from it regardless.
[XI] Surivere of Valefor
[XIV] Sir Surian Bedivere of Behemoth