Aren’t R’hllorism and magic fundamentally connected? Every red priest we’ve seen openly practices magic. They also seem to use magic as a selling point for the religion. Melisandre is a true believer, while some of her magic is intentional misdirection, she thinks and does real magic and attributes it to R’hllor. If today high ranking Scientologists came out and started performing real magic and predicting the future and performing miracles, you’d think they’d get a lot of converts from other religions. Like whether the religion is true can’t be proven from the books, but she is in an honest to god magic cult, and that seems like it should be appealing.
Fundamentally? No.
As I’ve discussed before, there are R’hllorite priests who know non-R’hlloric magic (like Melisandre or Benerro), there are R’hllorite priests who do not know non-R’hlloric magic who do R’hlloric magic (Thoros, Moqorro although he could be in the first camp), and there are people who are not R’hllorite priests and who don’t know non-R’hlloric magic who are able to spontaneously perform magic associated with R’hllor (Beric).
At the same time, there are plenty of examples of people who use the same kinds of magic without any associations to R’hllor: Dany sees fire-mages and meets shadow-binders in Qohor, there’s Bloodraven, etc.
Moreover, R’hllorism is not the only faith that is associated with magic – the greenseers of the Old Gods, the water-wizards of the Mother Rhoyne, the miracles attributed to the Seven or the Drowned God, the secret association between Valyrian steel, blood magic, and sacrifices to the Black Goat of Qohor, and so forth.