There is no way I will be able to fit the full rework into the 3000 character limit of a forum post, and I'm not spending any more time using HTML formatting than absolutely necessary. Please refer to [THIS] published Google Doc for the full rework, and throw any feedback or suggestions in the comments. I will make no mention of specific potencies, with some mentions of effect durations, because I am not a game designer, nor am I fully versed in game balancing. Think of this as the skeleton and muscle of a rework. The outline). THIS WILL REQUIRE A FULL REWORK OF THE ARC AND SMN CLASS/JOB QUESTLINES. (which unfortunately makes this a very unlikely thing to happen).

Abstract

Pet Problems
You are Carbie. Carbie is you. From a purely technical standpoint, you and your summons will be one entity. Summons are no longer pets, but an extension of you as a player. This should circumvent the pet delay issue by simply making your summon a proxy for abilities that the game considers you to be casting and simply making them the origin point for the animation and effects. This will give the illusion of being a pet-based job while hopefully not dealing with the delay drawback.
The Core
Multi-name gauge which goes by the general term Summon Gauge, starts as Elemental Gauge. Build gauge by landing Summon actions (including Carbie's), by landing Ruin while under the effects of Aethercharge, and through your Brand New DoT, Drain. Full gauge grants 1 stack of Elemental Fury, allowing you to summon one of your three elemental Carbie forms (Fire, Wind, Earth, as usual). Elemental Gauge becomes Primal Gauge. When full, summon the Primal related to your elemental form (Ifrit, Garuda, Titan). Once the Primal dissipates, return to Elemental Stance and gain one stack of Primal Aether (replacing the mostly useless Aetherflow gauge) and put the used summon on cooldown, repeat for the next Primal. Upon completing two Primal summons, based on the last element stored in your Primal Aether Guage, the corresponding summon button upgrades into an Improved Primal Summon (Ifrit > Ramuh, Garuda > Leviathan, Titan > Shiva).

During your Improved Primal Stance, the Summon Gauge becomes the Eikon Gauge. Once again, fill the gauge to transform your currently active summon action into an Eikon summon (Ramuh > [Read], Leviathan > [The], Shiva > [Doc]. During each Primal Summon, every 25 or 50% of the respective gauge you fill, you gain one use of a summon-specific ability, and each Eikon performs a unique attack when it dissipates.