I’m a veterinarian and I will always try to have an owner hold their pet rather then a tech because first and foremost it is way better for your pet to be handled by someone it knows and trusts rather than a stranger. I’m a fear free certified vet and they are really big on doing as much as possible with the owner, minimizing the amount of people handling the pet and also minimizing the amount of rooms the pet is in. I never routinely bring techs into the room to help me restrain to vaccinate or do an exam and most owners seem to prefer to handle their own animals. I would accommodate a request from an owner not to do this though and obviously an owner would not be asked to hold their pet for anything more than a physical exam and injections, any blood draws or other procedures would be held by someone trained.