One thing I want to throw into the mix is that if we're being honest, a pet class is not a pet class functionally so long as all it's pet does is the following thing:
* Have an external model.
* Autoattack a target or a group of targets.
* Use abilities either on CD or on command.
Because, at no loss of gameplay or output, this can be replaced with a stronger autoattack from the player + making the player use the abilities directly. In fact you could pretend either way, just based on who you assign as the originator of the damage on the server.
Two really really really important parts that have to happen to truly elicit a "pet class"-feel are:
* The pet must be vulnerable. It is your obligation as the pet controller to care for it / heal it / move it around danger / replace it as needed (specifics depend on class and game).
* The pet must be a significant portion of your total power, or ideally the majority. This means having your pet be healthy + attacking is far more important than you doing so. As a result the previous point gets priority, and since you provide a smaller portion of the output, you are "free" to do this, and focus mental time into it.
So at a very basic level, pets need to not be immune to AoEs, and in particular ground AoEs and effects. Moving them around these has to be a constant effort on the part of the controlling character. They also must be perceptible to full room AoEs (minor mitigation is alright, but not major one, or do it like in WoW where for a long time the Felhunter was the only pet with significant resistances) and the controller must have a way to invest time + resources into healing the pet and as per point #2 above, this is a damage gain to do compared to losing the pet.
That is to say, a pet controller's gameplay must be, well, "controlling the pet" for it to feel like a pet class. If the pet just exists on top of playing like any other caster there is, factually, no reason to have it in the first place. It's just window dressing, a DoT with an external model, might as well, well... neuter it. Which is what ~all MMORPGs do, as evolutions in both player class and boss encounter design gets too restricted by the nature of pet classes. However, I will say FFXIV is unique suited to complex pet classes compared to other games.
We already had/have jobs suited to being the DPS and healer pet classes (they never truly were as per the above, but eh, doable), and we know from previous work with the combat potato that tank pet classes can work in a combat system as slow and methodical as FFXIV. Even boss positioning could work!