Not to be picking holes in your coat, but the exact wording is "Congress"; shall make no law...etc." not "government".
There is a BIG difference.
The Congress of the United States is a single, established, limited body of representative governance of and for the United States.
OTOH, The "government" of the United States is a wide open term that can mean anything from the very highest office in the federal government, to the elected part-time dog catcher/mayor of Teenytinytown, N. Dakota (pop. 751).
Why I bother to bring this up -- other than risk making a complete nuisance of myself -- is b/c it is this phrase ("Congress shall make no law...") that is the bone of contention in the tug of war over Christmas scenes, Ten Commandments displays, etc.
To wit:
--One side states that this line in the Constitution is, in deed if not in fact, the "separation of Church and State".
--Whereas the other side points out that the line read that "Congress, shall make no law..." Congress. Not the state and/or local governments. Congress and only Congress shall make no law(s) establishing, etc.
Therefore, some argue, if the government of your town decides it will permit an organization to put up a manger display in the city park, it has every right to do so. B/c the Constitution specified, they say, that "Congress", not simply "the Government" shall make no laws respecting the establishment of religion.
To a certain extent I understand what you're trying to say, and I think it certainly valid that you corrected my 'government' to 'congress'. The government of the United States does refer to the federal process. I think it's a bit of a stretch to consider local and state offices as the government of the United States.
Now that may be academic, since everyone tries to extrapolate local rules based on that federal document.
The congress is the lawmaking body of the federal government. If people want to extrapolate meaning, they should only extrapolate that the government is not to establish. It is not to govern how and where religion is practiced.
If people in a local school board want to have the kids pray every fifteen minutes, it's not the fed's right to strike that down. If the feds think that it is, then they should rethink the provision of education in this nation, back out of it and let the local communities provide that education as it was always intended.
If someone doesn't like their children attending schools where prayers are offered, let them be the ones to start up schools on their own. Why should the religious be the ones who are outcasts?