While I have not seen every American vet (obviously), I have yet to see one properly administer inoculations.
Not one ever asks if the animal has been sick or had a fever in the last 30 days, whether the dog has had any other shots in the last 30 days, or holds the animal under supervision for the required time after the shot (can be several hours).
The only question ever asked is “will that be cash or charge”.
You’re absolutely right. No sick animal should get any vaccine. But if the owner doesn’t object, the vets will give it anyway. They sure do care about your pet, don’t they?