Summoner retains carbuncle from a lore perspective because Egi and Carbuncles are similar constructs. Likewise, from a lore perspective, Sch does not because Lily is her own living thing and not something constructed by the Scholar. Once again, with lore, the mere act of summoning and many spells used by Summoner require many on the spot calculations and math: that's why you're writing in your book when you summon. Gameplay wise, Arcanist and Summoner are exactly the same, while Scholar is the third wheel. In addition to that, Scholar currently shares a single action with Arcanist, which will be 0 if SE has their way: Resurrection.
You may want Scholar to be the one that goes on its own, but neither game play nor lore support such a thing.