But doesn't this argument also mean "Allowing influence of religious people over laws (e.g. allowing them to vote) is unconstitutional"?
You might think that you're being completely neutral when you apply your atheist worldview to political questions, and that it is only those weird out-groups that are biased with their desire for laws that your in-group opposes, but from their perspective you are being just as biased by your religious beliefs (i.e. your beliefs about religions).
If you really think that the First Amendment was written with the intent to invalidate any policy which can be supported by some interpretation of the Bible, then you haven't thought through the consequences of that.
That’s fine but it’s not what we’re talking about, which is what keeps the democratic coalition together. Opposition to Christianity having a public role isn’t it.
Furthermore, those who think the Bible should have influence on US laws fundamentally fail to understand the very basics of the first amendment of the constitution: "Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof."
Allowing influence of a particular religious sect over laws is exactly akin to a state religion, which is expressly unconstitutional.