Augustus
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I can't recall doing anything like that when I was an atheist.
I've never met an atheist who thinks 'because God said so' is a legitimate reason for them to believe something.
People probably take politics the way it works in democratic societies. If you're not active, chances are politics will turn on you and take away freedom to do something you like for some reason you don't agree with. Probably the abortion debate is most polarizing at that. Evolution is hardly a matter of debate here, but seems big in the US.
The degree it is present
And what of atheists who think religion should play a role in politics? There's a rather infamous local low-brow atheist who is against Islam and has tirades against other atheists, LGBT, liberal Christianity and "cultural marxism." Well actually he's in "exile", government is trying to extradite him from Monaco. He's good buddies with fundamentalists, who somehow ignore he's an atheist.
They can do that if they want. It doesn't stop other atheist thinking differently.
Even though I'm now monotheist I don't feel compulsion to go to Church either, even though the bells tolled some hours ago. That's because it's not my religion and I don't agree with the theology. I don't think they present a path to God at all there. If someone gets there, it's by no help of that church.
We all have different reason for things.
If you asked 100 atheists 'why don't you go to church/mosque/temple?' I'm sure quite a few would say 'because I don't believe in god(s)'
I think a lot of it is due to the environment where you live. If you were Scandinavian it would probably be something of little consequence, unless you didn't like paying money to the church. I think being a vocal theist would have more consequences, since people tend to keep their beliefs and mostly politics a private matter.
The more religious the society, the more you are forced to think about these things. Ultimately though, the same beliefs are present with the same potential consequence. Anyway, it's not like Europeans don't pay attention to world affairs or look on in bemusement at US fundamentalists.