As an atheist, I don't think they are fully running the show. I do think that religious groups have far too much influence though.
I also don't think the average christian realizes how many christians in america are in a proper death cult and not just a weekly golden rule club. They call themselves the same word but they do not believe the same. I was raised in a church that talked about revelations nearly every sunday, my wife was raised in a church and didn't know there was a book in the bible called revelations. Not the same at all.
Another point is a point of view problem.
It is hard for a christian to see the effects they have because it is normalizing from your perspective.
Kinda like when Elon says free speech he means free to agree with him. No one who agrees with him will be put off by him censoring content he disagrees with. They just notice more speech they agree with and assume that means more freedom because clearly any free and sane person would think like them and talk like them.
America's freedom of religion has always really been a freedom to pick your branch of christianity. Don't believe me? Go look up the study where more american people said they would rather leave their kid with a pedophile than an atheist. How welcome and free do you think that makes me feel?
It is the non christians who are hassled by things like prayer in schools or the 10 commandments in courthouses. They seem like trivial non issues to believers until you flip the frame for them. Would you want a baphomet statue in the school or at the courthouse you have to explain to your kids that you are trying to raise as good christians? Your prayer before class is the same for me and my atheist kids.
10 commandments in courthouses started as a marketing stunt for the movie but the christians lose their minds over it. What would a jesus who flipped the tables of the money lenders think of their obsession with a movie marketing stunt over the substance of helping their neighbors? (I'm the devil he warned you about knowing the bible and quoting it for his own purposes)