Many of the founding fathers were deists. For example, Jefferson believe that Jesus was NOT God but a very moral and good man. His Jefferson Bible promoted Jesus morality while excluding his divinity.
Thomas Paine certainly did not believe in Christianity. And some, like Washington, attended Unitarian churches which most today would not consider to be very religious.
So, no, I don’t think religion has a role in the government, but morals—whether of secular or religious origin—do have a role.
Indeed, we now have a population largely poisoned by pernicious lies, bigotry and hatred. There is little good faith left.