Your fan theory that all the wizards are just pretending is interesting, but unsupported by canon.
First, that's not my theory.
Second, your theory that ASOAIF gods provide magical powers is not supported by canon.
It is at best left ambiguous.
The majority of which are one-line mentions in the books for worldbuilding flavour purposes.
You said - "
one major religion in the setting that lacks any supernatural power".
Since we are discussing the religions in the setting, the fact that followers of Essosi faiths are not in the middle of main narrative is immaterial. They exist, they are important in the setting and have no magical underpinning.
Red Priests resurrecting the dead and literally casting fireball spells isn't ambiguous, and neither are what the Faceless Men do.
If Red Priests as a class could resurrect people and cast fireballs, they would rule the world by now.
If anything, Thoros (who grew up in the temple surrounded by Red Priests) is mighty surprised by his newly found ability to resurrect people. Which would not happen if Red Priests could just do these kind of things.
We have two Red Priests out of however many thousands out there who can do tricks like that, so there is no reason to presume that it's their status as Red Priests that is responsible for the magic. Granted, Mel is on a power trip and Thoros is on guilt trip, but their desire to have their every action justified by God a proof does not make.
Ditto for Faceless Men. Kindly man says so directly, for crying out loud: "...sorcerers use glamors, weaving light and shadow and desire to make illusions that trick the eye..."
Faceless Men learn magic that is available to skilled sorcerers, not just people following their god. Undying use glamors. It is no more divine than learning how to operate an assault rifle.
It's heavily implied that there is something supernatural and very nasty about that religion, not the least with Patchface.
It is not "heavily implied" - the connection is barely existent and is primarily a fan theory with very little basis in canon.
Debatable? They not only have magic, but it's all but stated that their afterlife about merging with nature is real.
Yes, debatable, because of millions of Old God's faithful only a small handful demonstrates an sort of magical ability. And most of those reside in a land beyond the Wall, where magic is stronger and evil fairies roam around.
So again, you attribute to the Old Gods what is by no means proven to be their doing.
To discuss the remaining ones:
Great Stallion: Foresight
Considering that that bit of foresight has failed like a wet fart, that's hardly in argument in your favor. The Stallion is not mounting the world by the virtue of being dead.
Horsemen are just as bereft of supernatural powers as the Faith.
Mother Rhoyne: Hydromancy,
The only mention of Hydromancy by MR's followers dates back hundreds if not thousands of years ago. By this metric, Faith of the Seven is magical because similarly distant reports tell of Gods roaming Andalos and pulling stars to crown Hugor. Ditto for Drowned Christ.