It’s the owner who’s at fault here, not the dog. Some dogs are reactive (mine can be) and it’s the owner’s obligation to manage the situation. I know when I go to the vet I have no more than 2 with me and I keep them on a short leash (it’s a 6 ft leash but I wrap it shorter around my hand) and I find a quiet, out of the way spot for us to wait. I then am vigilant about others approaching. The vet staff should be policing this kind of behavior for everyone’s sake.
If there’s not enough room for us to be at least somewhat separated, we wait outside or ask to be put in the room right away.