Note also (pby) that the establishment and free exercise clauses share the noun "religion". The word is only used in the establishment clause and simply referred to in the free exercise clause: "or prohibiting the free exercise thereof".
Therefore if you construct the establishment clause such that "religion" must be understood to mean something like "state religion" or "national religion," then the government would only be prevented from prohibiting the free exercise of "state religion" or of a national church. But this construction is (one would hope) obviously absurd.
If you are right, then among other evidences:
Why does Congress open in prayer?
Why does Congress have a Chaplain?
Why did, until recently, the United States give a Bible to members of the military?
Why was the Bible used as a textbook in public schools?
Why are there direct references to God and religion on federal buildings, federal memorials and federal statues? Why did a three judge panel just decide for the public display of the Ten Commandments and other religious documents and in so doing stated that, given history, no reasonable person would agree with the ACLU's position on the alleged wall of separation?
You wrote: "then the government would only be prevented from prohibiting the free exercise of "state religion" or of a national church. But this construction is (one would hope) obviously absurd."
Reply:
Why is this absurd?
Churches have always tried to capture the power of government to promote their self-interests (tax-exemption, for one).
Is there a single word or phrase in the Constitution that says government has a positive obligation to promote religion? Obviously, there is not such a phrase. Religious beliefs and church affiliations were obviously meant to be left to the private sphere, the family and individual conscience, and not to public displays of religiosity and pretended piety.