Who is in charge of decided what ideas and morals are false? Is there a secret committee that only atheists are members of?

Nobody is/everybody is, but when you are basing your morality, what you think is a good or a bad decision, that has no grounding in reality, you are far more likely to make the bad decisions. It's not so much about what is moral, it's about doing what is best, rather than deciding everything based on some abstract concept of "this is right no matter what" and "this is wrong on matter what".

I do feel that the point of the fight over religion, God, etc. is that no one has any sort of proof one way or another.

You cannot find proof of something not existing. You can find things that contradict a thing exist, such as there being zero signs of it existing, explanations for everything without needing some deity to fill in the games, etc, but you cannot find disproof of a god. It's literally impossible.

It is impossible to run experiments on a point of view.

It is, however, very possible to run experiments on how prayer does nothing, on how the sheer volume of the claims made in the bible have been proven false, and so on. Yeah, it's impossible to run experiments to disprove religions, but what does that say exactly? Look at any other claim, and look at the baseline for people to regard you as sane when making it. You need proof, you need evidence, you need backing.

Religion has none.