Joseph:
Your error here, among others, is that you are pumping the hitting thing as the preferred method of intervention. Your primary source, which I believe you are abusing, is that חושך שבטו שונא בנו a recommendation. That is clearly inaccurate. Rather, it is that the total withholding is problematic. The second half of the posuk, unknown to many but still present in all present publications of Sefer Mishlei, is ואוהבו שיחרו מוסר. You may consult again with Rav Wolbe ZT”L to see the simple explanation. Discipline is NOT the primary approach, and several of us here in the CR have presented our simple explanations. It is sad that your approach is to begin by proclaiming that this spanking thing is the primary tool, and that you now need to find a way to justify it.
Many of us reviewed your citations. NONE of them push the hitting agenda as the preferred tool in chinuch. You confuse the עיקר and the טפל. It is not right to attribute this to any of the resources you cited. Try some that I have cited multiple times, and approach the question with an open mind. Let the Gedolei Yisroel answer the question.