It simply prevents our federal and state governments from creating a state church and there is a difference. If it were intended to completely divorce religion from government it would say so. The document is written in plain English. To take it much further is simply over extending the words to suit our current whim. The second problem is that as we've backed down this path making sure the state doesn't found a defacto church, we've run squarely into the next clause of the First Amendment where the state is prohibited from interfering with the free exercise of religion.
Any government regulation is an infringement on the right to freely exercise a person's religion.
If we used the same logic that has been used to extend the anti-establishment argument, we would find ourselves with government being unable to create order in any situation where religious practice was claimed. For example, school graduation could become a wild melee of competing religious factions and school boards and their representatives would have no ability to create order at the event. We haven't succumbed to this overreaching interpretation yet. However, the Supreme Court has chosen to favor the anti-establishment clause over the free exercise clause. To do so is wrong.