Also, that'd require a rework of the entire Ops groups system, to allow a pet or companion out in an Ops group.
Actually, there's already an existing mechanism used by the game that could be co-opted by a pet class/AC without interacting with the companion system *or* being impacted by ops groups. Most people just don't know or remember it's there bec/se it's used once and only by a specific subset of characters: the Biochem benefit on Athiss. It throws a unique buff on the player that summons an Akk Dog that follows you around just like an extra companion. The only way that it's different is that you don't get a nice little UI set up to see all of its information and, in its current implementation, it's a 10 minute buff that falls off immediately if/when said pet dies.
It wouldn't be that difficult to coopt said mechanism, add a couple of additional class abilities that mess with your pet's threat table (one that generates a functional infinite amount of threat on a target to get your companion to attack that target; another that zeros out all threat to get them to stop attacking) and have an entirely mechanically viable pet functionality for a class. The pet summons themselves could be the "stances" of the class (one summons an mdps pet, another summons an rdps, and a third summons a pseudo-tank, support, or healer) that are identical to the previously mentioned buff with the exception of indefinite durations.
The problem with a pet class (or any new class) isn't in the mechanical viability of it. It's in the developmental costs associated with writing, voicing, and, potentially, creating an entirely new starting planet.