Actually, your statement is not correct. The Constitution did not establish the United States; it established the general government which was to serve as the agent of the states which were the principals. Do not confuse a particular polity of a country with the social order of the country which was historically from its European roots Christian. In the 18th century, as a little further corrective, the word "religion," basically a new word in the context in which we use it today, meant "church" or "particular faith," i.e. Baptist, Catholic, etc. Our modern notion of "religion" as the amorphous ether lodging in the individual person was basically alien to the 18th century. If you read the ratification documents, which are the documents which count since it was the ratification conventions which gave the Constitution any authority it may have or have had, you will see that they understood the phrase to mean that Congress could not establish a church nor could Congress prohibit a state from doing so. In fact, at the time, there were a number of state churches. As a matter of rational exercise pertaining to "religion" as something different from "church," do you really think that anyone perceived a danger that a sub-committee of Congress would one day sit down and conjure up some "religion" ex nihilo and foist it on the American people? No, that is no logical.
You may not want Christianity to have any role in the American social order or in its polity today. That's fine. But let's not do a little creative re-writing of history. Such things are better left to folks like Joe Stalin.