The only bona fide centuries-old (actually, millennia-old) religious institution I can think of is the Catholic Church. All the others are largely religious traditions, more or less, that are centuries-old, but not continuously operating institutions. There is nothing here that suggests these guys had any love for the Church (certainly, their conspiratorial mode would not sit well with the rational spirit of Catholicism). Just because someone says "Jesus!" or "Satan!" doesn't make him a Catholic, or even a Christian or a Muslim, for that matter. For all we know, these guys thought Pope Francis is a reptilian who drinks the blood of altar boys who've recovered from COVID, or at least the usual array of Protestant conspiracy theories about the papacy.
(I might add that calling the Catholic Church shady/mysterious is giving too much credit to whatever malicious elements exist within it, similar to how conspiracy theories in general assume a scale and degree of coordination, competence, and secrecy that is practically impossible. As the joke goes, if a business were run this way, it would not last two weeks, let alone two millennia.)
(I might add that calling the Catholic Church shady/mysterious is giving too much credit to whatever malicious elements exist within it, similar to how conspiracy theories in general assume a scale and degree of coordination, competence, and secrecy that is practically impossible. As the joke goes, if a business were run this way, it would not last two weeks, let alone two millennia.)