First of all I did not offer a dichotomy false or otherwise. You have just offered a generalization of the opinion of all atheists, while I was offering an interpretation of the opinion of one particularly litigious atheist in the article.
Second, the idea of the government to "allow all religions equally" presumes a government above all religions answering to a system of ethics which is above religion...or more to the point: above OTHER religions...and conforming to secular religion. In other words, you are saying atheists want government to keep each of the lesser religions (the ones that have god or gods or something super natural in them) on a leash and keep them out of public areas. I don't think all atheists are as dogmatically committed to subjugating other religions beneath secularism as you make them out to be.
What our constitution says is that Congress should not make laws about religion, not that Congress allows religions equally.