I don't know Strauss. I agree with his quotations you posted, although I'm not so sure about the line drawn at the Christian empire. Niccolo certainly distrusted Rome. And he did draw his study from Livy. I think that Strauss' connection of the two is dubious and leads no where, anyway.
Niccolo was concerned with what works. What's more conservative than that?
A good article on Strauss is
HERE. It also deals with the author's opinion of where in importance in Strauss's thinking Machiavelli stood.
A good trade paperback available a fairly reasonible cost is the History of Political Philosophy editied by Leo Strauss and Joseph Cropsey. It is a mammoth overview of just about every philospher that was of political importance.