It may be enlightening to discuss James Madison's interpretation on the establishment clause specifically, since he was the one who proposed it to congress, with the rest of the bill of rights.
He discusses it in great detail in Detached Memoranda, which is
available online.
Here are a few parts I want to point out.
In strictness the answer on both points must be in the negative. The Constitution of the U. S. forbids everything like an establishment of a national religion. The law appointing Chaplains establishes a religious worship for the national representatives, to be performed by Ministers of religion, elected by a majority of them; and these are to be paid out of the national taxes. Does not this involve the principle of a national establishment, applicable to a provision for a religious worship for the Constituent as well as of the representative Body, approved by the majority, and conducted by Ministers of religion paid by the entire nation.
I think the bolded part is important, what he is saying is the constitution does more than just forbid the establishment of a state religion, it goes much further, as he puts it, the establishment clause forbids anything like an establishment of religion.
You can also see the intent by reading some of the earlier drafts of the first amendment
seen here. Many of the earlier drafts simply forbid the establishment of a state religion, the final draft, added the word "respecting", indicating that this was intended to have a broader prohibition.
Another interesting point by Madision which should shed some light on things like a "National Day of Prayer". I found this a few paragraphs down from the last quote:
Religious proclamations by the Executive recommending thanksgivings & fasts are shoots from the same root with the legislative acts reviewed.
Altho' recommendations only, they imply a religious agency, making no part of the trust delegated to political rulers.
He goes on from there.