After the Constitution went into effect, the original States ratified the Bill of Rights - the First Ten Amendments - which were specifically intended to limit the power of the new Federal government.
The First Amendment begins:
"CONGRESS shall make no law respecting an establishment of religion OR PROHIBITING THE FREE EXERCISE THEREOF ..."
Let us examine the meaning of each word and phrase in the First Amendment:
The word "Congress" meant the "Federal" Congress - the one branch out of the three which makes laws. Article One: "All legislative powers ... shall be vested in a Congress ... Bills ... shall originate in the House."
SHALL MAKE NO LAW
"Shall make no law" meant the Federal Congress could not introduce, debate, vote on or send to the President any bill respecting an establishment of religion.
The founders did not foresee that Federal Courts would usurp power and effectively make laws from the bench, or that Presidents would effectively make laws through Executive Orders and regulations.
Had they anticipated that this would happen, they likely would have worded the phrase: "Congress, the Courts and the President, shall make no law ..."
The word "respecting" meant "concerning" or "pertaining to." It was simply telling the Federal government to keep its "HANDS OFF" all religious issues.
When the topic of religion came before the Federal government, the correct constitutional response was to be nothing, as the Federal government was not given jurisdiction on that issue - neither for nor against.
Religion was under each individual States' jurisdiction.
AN ESTABLISHMENT OF RELIGION
"Establishment" did not mean "acknowledgment." It did not mean the mere mentioning of God, Judeo-Christian beliefs, or prayers.
Establishment was a clearly understood term. as nearly every country in Europe, as well as most of the colonies, had establishments of religion.
It meant that one particular Christian denomination had its organization, hierarchy and staff structure recognized exclusively by the government in preference to all other Christian denominations.