You are entirely correct. The quotes don't support the heading (Against Religion in Public Life:) at all. They object to government interfering in religion, not to religion influencing government. Most of the founders found religion to be a highly salutary influence on government.
I'd be satisfied with the situation as it was in the '50's when I was growing up. That was before atheists convinced the Supreme Court that they were allergic to the mention of God and we listened to Kate Smith belt "God Bless America" on TV every night without any controversy.