A morality that recognizes that my rights end where they infringe on yours, yes. That is a necessary part of the law, which exists to protect our rights.
A morality about who you can sleep with, what god you worship, what you can drink or eat, because my god says so, no. That would not be consistent with a free society, which is what the founders were trying to establish.
That they believed only a moral people could maintain a free society is one thing. Of course that's true. But legislating personal choices to conform with your particular flavor of religious authority is another.
A people can be moral without laws telling them how to do it. They are different concepts. Not everything should be subject to law. They understood that distinction. It seems to be pretty rare these days.