Can't have them finding out that the WT worldview and lifestyle might be making the problem worse, now, can we?
While not exactly on point, I always felt that keeping mental health problems in-house was motivated by the same logic found in 1 Corinthians Chapter 6:
Does any one of you who has a dispute with another dare to go to court before unrighteous men, and not before the holy ones? Or do you not know that the holy ones will judge the world? And if the world is to be judged by you, are you not competent to try very trivial matters? Do you not know that we will judge angels? Then why not matters of this life? If, then, you do have matters of this life to be tried, is it the men looked down on in the congregation whom you assign as judges? I am speaking to move you to shame. Is there not one wise man among you who is able to judge between his brothers? Instead, brother goes to court against brother, and before unbelievers at that! Really, it is already a defeat for you when you have lawsuits with one another. Why not rather let yourselves be wronged? Why do you not rather let yourselves be defrauded? Instead, you wrong and defraud, and your brothers at that!