EDIT: When a peasant revolution starts, it needs some theory behind it to get real results. I'm not saying "there must be a vanguard party leading the revolution and giving orders", but there needs to be some sort of shared idea and collective demands and values. Even if you don't want to overthrow the state, you need to know what you want so that the movement keeps going until you get it. Establishing this mid-revolution is crazily hard. Westeros doesn't have concepts like socialism or Enlightenment human rights that were the ideological backbone of many successful IRL revolutions, and the peasantry doesn't even have a tribal/national identity (or clear leaders) to rally around when they fight the Lords. Therefore they rally around the Faith, which strictly speaking does have a bunch of rules that basically boil down to "human rights + don't be a dick to each other". Sure nobody applies them to the country's leaders normally, but they're there in the holy texts and have divine authority backing them, so in times of strife they're a perfect thing for the peasantry to unite behind. Indeed, IMO it's significant that the parts of the Faith's theology we know the most about are the commandments the sparrows keep invoking, and not any commandments the lords of the land could have been using to justify whatever they do (like IRL medieval nobility often did).
I agree with all of this, but I have to say that is also why the Faith aspects of the canon plot are so unsatisfying. Peasant rebellions are by their nature doomed, they are not revolutions, they do no transform society. Kill a few nobles, burn a few castles, but in the end the nobles are still going to noble and the peasants are still going to work the fields. It's practically impossible to root for them in canon because they are as depressing as they are directionless. GRRM wanted to show the struggles of ordinary people, the horrors of war, but by dressing them in the guise of religious fanaticism he basically made his underdogs unlikable, or at least less likable than they should have been. when the High Sparrow makes Cersei do her 'walk of shame' it's a lot easier to sympathize with Cersei than the Sparrow because that is a barbaric thing to do to anyone by modern sensibilities. So in the end we are still left rooting for Arya, for Jon and for Dany, nobles all not for the peasants and their priestly leaders.