It looks as though the exoteric public Strauss stands up for truth and philosophy against nihilism, historicism, and relativism. While the esoteric private Strauss may be saying that the truth is ugly or empty and that most people can't handle it. Some may only have heard the public Strauss, not the private one. And for those who've heard the private teaching, it's not clear what the next step would be. In any case, the public Strauss is clear in his condemnation of modern relativism, but the private Strauss looks more like a teacher or a Socratic questioner who lays forth the dilemma than a dogmatic or doctrinal philosopher who presumes to answer questions authoritatively.
Some of this ambivalence has to do with Socrates. Philosophy is to be valued. It is of the highest importance, yet it may be destructive of society. And some ambivalence doubtless relates to Germany in the interwar period: to the position of the thinker between those who can't rise to assert anything definitively and defend it and those who unquestioningly pursue a single goat regardless of its consequences or other considerations. Rather than viewing Strauss as the problem or the solution, he looks like another instance of the dilemmas human beings face in times of conflict.
Yes, x. I concur with your last. But this is ambivalence of a unique kind. It is one that admits the independent and extra-anthropological existence of truth. Yet it doesn't deny the human participation in it. The implications of this stance are profound in the area of law.