No. The Framers were practical men, and they recognized that different states had different dominant denominations in them. Let us be clear, that the founders weren't using the term "religion" in the context which we modern folk use it. They were referring to differing denominations of Christianity, and perhaps a little bit to Judaism, though I don't know of any specific support for that.
As I said, the Framers were practical men, and they knew that if they designated a particular denomination as the "official" religion of the United States, it would blow apart the coalition. The Quakers of Pennsylvania, and the Puritans of Massachusetts, and the Anglicans of Virginia and the Catholics of New York would all be at each other's throats over this issue if a particular denomination was selected.
The Founders wisely chose to avoid this conflict because they needed the entire coalition to resist the efforts of England to re-subjugate them again.
But that the Nation would be Christian was implicit and a given. The US Constitution itself refers to "Our Lord" meaning Jesus. The Articles of Confederation also refer to the Christian God, as does the Declaration of Independence. Christianity was incorporated into many of the State Constitutions of the time, as well.
Yes. I’m sure they also wanted to avoid a situation in designating a national Christian denomination where the President would serve as “Head of the Church”, as the King of England did. So many people today don’t have a clue about the Founders intent.