The inauguration is an official ceremony of the state that should have nothing to do with religion.
And it doesn't. It has the president-elect swear to uphold the Constitution. The fact that it includes a religious prop does not make it a religious ceremony.
I never said the ceremony was religious. The Bible is. It is specific to one faith. I am saying that it should be devoid of religion.
The idea of separation of church and state is that people have a choice, and a government endorsement of atheism is just as bad as it endorsing Christianity or Judaism or whatever.
The lack of endorsement of faith or religion is not endorsement of atheism. It's an endorsement of nothing.