If pets do what you say, and assuming you also do not want them to be vulnerable to AoE damage, then there is no reason to have them as pets.

That is, the devs could "fake" them having an entry in the group window, but they don't need to exist as selectable entities at all. They're just external DoTs at that point, since they have no failure point that isn't based on the player controlling them, just like missing a DoT-reapplication or so.

I'm not against it mind you, visually and theme-wise it's a neat choice for a lot of classes, but it is weird for players to always want "more pets" but also not want separate entities around the battlefield which is kinda what pets ought to be, no?