She wasn’t a fanatic, she was cold and calculated.

Eh. She was detached (often because she was drunk) and ruthless. But she was also obsessed with the Targaryans, initially because Rhaeger rejected her for Lyanna Stark but eventually because she needed to justify incest with her brother. She was absolutely a fanatic. A racial purest, obsessed with the perpetuation of her bloodline. The "at least she was a loving mother" line others attributed to her was far more about her fixation on continuing a permanent line of Lannisters.

The “fanatics” were the religious faction that was gaining power.

The Sparrows were a branch of The Church of the Seven composed of impoverished working people ravaged by the endless wars between the Houses. They were certainly orthodox in their beliefs. But the "fanaticism" tended to be described as disrespect for the aristocracy and militant organization outside of the major Houses.

Hell, the most notable aspect of the Sparrow leadership was that it was not beholden to the throne. The High Septon under King Barathon was as much a party to the intrigue and debauchery as anyone else in the council, and this passed as normal behavior. He wasn't above having people tortured or executed for apostasy. He just knew when to look the other way as it served the King. Breaking from the corrupt practices of the captured church was only labeled "fanatical" in so far as it ran afoul of the demands of a weakened central government.

Stannis and the Red Lady were significantly more fanatical in practice, given the number of people they burned alive and the amount of out-and-out witchcraft they performed. The Sparrows were just factional in so far as they refused to "bend the knee" in the same way as their predecessors.