What is wrong is that if we have ruled out physical issues, then it's either rider error or training or balance--most likely balance. It can be rider balance or horse balance.
If it is rider error--the rider is asking for the canter depart incorrectly--then it is unlikely that the horse understands why he is being forced to canter along on the wrong lead. Further, it is unlikely that the rider knows how to properly balance the horse. The solution here is not to make the horse tired, sore, resentful of the canter, etc. but to take lessons on how to get the depart. That means learning how to balance and set up for it as well as the aids to ask for the canter.
The other reason the horse is picking up the wrong lead is balance. The horse is not balanced enough to pick up the correct lead, either in his own body or because of an unbalanced rider. If the latter, the solution is, again, lessons. These will help the rider learn where he is unbalancing the horse. If the former, the horse is not balanced enough to sustain a counter canter through a turn and therefore cannot help but fall on the shoulder and onto the forehand in the canter. How is this productive in schooling a green, unbalanced horse?
There is a reason the counter canter doesn't show up on dressage tests until the higher levels. The balance required for a PROPER counter canter--and not just going around on the opposite lead--is pretty high. Lengthenings in the trot and canter and lateral work make appearances before counter canter.