So, I'm assuming that you also have some qualms about how Facebook or Google does business?
Does that mean you refuse to use Facebook or Google?
By using commercial services, you are actually supporting them financially. You can go to most any church without donating anything at all.
Interesting. In my (predominantly Catholic) country, people do that for weddings and funerals, it's widely accepted and in fact it would look radical to stay out of a wedding for not believing. But I don't think non-believers going to "regular" mass is a common thing, I think it would be seen as odd and probably not very welcome by the believers.
Also, one of the main prayers said in Catholic mass starts with "I believe in God, the Father Almighty". So consciously being in a group that says that looks literally like lying to me, even if it's a white lie.
[1] Incidentally, it's not that I'm not open to changing my beliefs, it's that (in my experience) believers are aggressively awful at converting nonbelievers. In many cases they don't even accept that it's possible to be a nonbeliever, immediately falling into "angry at God" tropes.
Given a prayer to a god one doesn't believe in feels dishonest for some, as does nodding along and letting people assume you share their beliefs when you don't. The foundation for church is supposed to be a shared theological belief, so it's not too difficult to understand why some might be uncomfortable joining when they know that they don't share the belief and have to go through certain motions to conceal this fact from their brethren.