I am sure this is serious research in a rather novel area. An animal model for stuttering is not so far fetched, if you believe stuttering is not the result of external environmental factors. If one studies brain development it can be seen that except for the neocortex human and animal development are very similiar.
Complexity is the primary difference between the brains of humans and animals.
A good text showing this is Biological Psychology by James W. Kalat. I think question to answer is whether or not the human behavior of stuttering arises from structures in the CNS which are unique to humans or are shared by both humans and animals. The question as to whether the zebra finch displaying stuttering like behaviors experiences depression, isolation and other emotional traumas because of how normal zebra finches treat the stuttering zebra finches is irrevalent. The emotional abuse experienced by human stutters is the result of ignorance and stupidity, however the emotional trauma is the result of and not the cause of stuttering. So questions about mating and social standing of zebra finches is not applicable because they would be result of and not the cause of the stuttering like behaviors seen in the zebra finch.