I think the experimental method seems right, but the premise is maybe wrong (it's not human history, it's Roman/Western history). These turnings as they are described, I believe, do not apply equally across civilizations.
In Democratic/Oligarchical civilizations (Roman, British, French, American) these "turnings" matter quite a bit more. Why? Because we're democracies! The public and its opinion is the pulse of the civilization, institutions depend on it. There's no Mandate of Heaven or concept of an eternal Middle Kingdom. We've bounced between demands for rapid change (oligarchs, kings, emperors) and returning power to the people. That ebb and flow, the "history rhymes" seems to me to be a reflection of the population aging, changing its opinion, and building and rebuilding over time.
In other civilizations, (Chinese, Persian, Japanese, Slavic, ... others) I'm not so sure the interaction between demographic changes, civilizational values, and governmental systems respond in quite the same way.