A fourth option: Another player runs the pet as his or her PC.- the DM has full control, the player interacts with it via roleplay (in the same way as with NPCs) and possibly some occasional check
- the DM lets the player have some limited control (e.g. can give combat commands but not choose the specifics) but the DM can always override it
- player has full control
This would, of course, require another player to be willing to do this, but I think that might be surprisingly common. It would also require a "pet class" with enough depth to work as a full-fledged character. However, it completely solves the action-economy problem. As a bonus, it encourages party cohesion.
Edit: Now that I think about it, the solution is not a "pet class" but a "pet race." Then you have the full array of class options available, which provides all the depth you're likely to need.
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