Your stock market sine curve analogy would show a 600 year, and continuing, decline.
As for the papacy, no I would say it depended more on the great prestige of the Roman Church from the beginning, because Rome was the metropolis of the Empire, and because of the presence of the tombs of Peter and Paul. Conjoined, the association of the two most prominent New Testament figures with Rome, is a firm basis for the claim of its bishop to be the vicar of Christ. As for the present standing of the papacy, think back two hundred years, or even back to 1870. By imprisoning the papacy. so to speak, Napoleon freed it of the chains that had bound it under the national monarchs. By stripping it of the last prestige as a secular prince, the Italians allowed him to exercise ecclesial authority freely. So freely, that national Church leaders often chafed under it. One object of many at Vatican II was get out from under the Curia. Theywere helped by the weakness of Pope Paul VI, a liberal pope, but even he was a disappointment. Humanae Vitae was more than just a" re fudiation" of sexual teaching that were being pushed by the "progressives" in the Church,it was a bombshell. Then along came John Paul II and now Benedict XVI. I am sure many liberals do think of the papacy asa rock--the kind that smashes their boats.