"Is there any reason why religious people (off whatever religion) can't found a completely secular institution?"

No. And that's part of what I've been trying to get at -- they did. And they weren't ashamed of the heritage that led them to that grand endeavor (as robomatic so eloquently stated). The Christianity of that era was instrumental in their founding of the secular US government -- and they didn't see the same kind of division that is now being demanded. They didn't develop the secular government as anti-religious or religion-free (as regarding informing its participator's moraity). In fact, they seemed to operate on the assumption of understanding the integral and necessary part that religion plays in informing morality.