In reading your response, I wonder if you were given the opportunity to study world and american history in school, or was it just another watered down curriculum that offered it as an elective.
First. They were reading the "Declaration of Independence", the reasons for rebelling against England. It was not a document for establishing a nation.
Second. Before the "Constitution" was the "Articles of Confederation", a document which loosely bound the individual colonies -which were in effect separate countries-. However, those colony/countries did in fact force religion upon any one who wished to settle within their borders.
Third. The "Constitution" was drawn up after the "Articles" failed. This all happened well after the "Declaration" in 1776. The other documents came in 1783 and 1789. The main thrust of the "Constitution" was to create one nation with a weak non intrusive federal government. The 1st Amendment which was adaopted later was to protect your right to freely practice any religion you wish without the government telling which one, in addition to other rights.
Finally. If a TV station wishes to air what it did, and you don't like it, you are free to mute or turn it off. That is your right.