You think a city that lives “by-the-Shadow” where the very stones try to devour light, wouldn’t be attracted to a religion that worships the Light and believes that “the night is dark and full of terror”?

Moreover, Melisandre says:
“In ancient books of Asshai it is written that there will come a day after a long summer when the stars bleed and the cold breath of darkness falls heavy on the world. In this dread hour a warrior shall draw from the fire a burning sword. And that sword shall be Lightbringer, the Red Sword of Heroes, and he who clasps it shall be Azor Ahai come again, and the darkness shall flee before him.“
That suggests that Asshai was one of the earliest centers of R’hllorism.