It is still not clear to me where the incorrect explanation is. Seems to work to me.
The angle at which total internal reflection occurs is called the Brewster angle, are you attempting to say that this is incorrectly handled? I think it is correctly handled.
Perhaps you are reading different books then I am. You tone seems to indicate that you have read them all. Is this possible? Or are you making some assumptions?
Rereading your post, and my text, it occurs to me that while you are impressed with it, you are ignoring it. Please read, and make an effort to understand the development. If you are unable to understand the approach, then please do not continue your criticisms of the general understanding, when it is YOU who is having the trouble.