I think politics and religion are sort of special targets for stereotyping.
If you are black, like country music, or are a gamer, you don’t have some official manifesto or doctrine people can point too and go “Aha! You gamers are all misogynist who marginalize women, it says right here!” If Toby Keith says something, people don’t automatically think that the rest of the people in country music and their fans have the same view.
With religion and politics, they do have official doctrines. They have a hierarchy with people in charge making statements. These people are the face of that organization. But the problem is people belong to this groups but not always agree with everything the organization stands for. This is because these organizations evolve and change as time goes on. Leaders change and the direction of an organization can change. Some organizations listen to their members more than others. People continue to support an organization they may not agree with 100% because they feel like it is the best choice out there, or they are actively working to change it.
But I don’t think you can point to these official statements and use it to justify a stereotype, if there is a significant population that doesn’t meet that. Just like I can’t wave a survey of girls that showed 89% of them chose pink as their favorite color and justify the stereotype that girls like pink.