I trained with a Korean in the past who used a small stick in such a manner a couple of decades ago because that is the way he was taught, but even he began to stop hitting students here in America. The purpose behind this kind of teaching is not that it gets the job done better, or teaches faster. It is more about checking attitude, pushing the student's "buttons," and testing the student's ability to take some tough treatment and not lose their focus, concentration, or lose their "cool," and get upset.
I use a kicking pad, or padded blocking bat in a similar manner, however I am more gentle with the children than with adults, and it depends on their rank. The contract that I have adult students, and parents sign when they join, specifically addresses the issue of physical reprimand. So long as I don't inflict serious injury, training for adults and high ranks can be tough, and they know that. I always explain to the students why I might strike them with a pad, and that it is not out of meaness or bad attitude on my part. If they can't handle some aggression in the class, they won't be prepared for an attacker in the street.
I also use a short wooden stick (a bo staff) for adjustments in foot, leg, hip, and shoulder alignment, but I don't strike the students with that. If they were fully padded, I might use a shanai, or such on the adults. The problem comes when an amateur instructor sees this done, and imitates it without understanding it, or without explaining to students (and parents) why it is done, or they do it out of control. You don't strike students in anger, or to punnish them, or to hurt them, but you might strike them with control when you spar, and you may choose to strike with a pad to test their ability to deal with strikes.
If it is used, it is something that must be explained properly, and done professionally for the right reasons, at the appropriate times, with proper control, and with consideration for the age, and rank of the student. If an instructor does not feel that he or she understands the purpose, benefit, or your ability to apply this type of teaching properly, then don't do it.
CM D. J. Eisenhart