You make a lot of sense. Who the heck would hold Lincoln to the standards of modern evidence? And who the heck would discount the quality of his message based on that? Its weird. His message was good and true. All people are equal. Lets not water it down with ridiculous semantics.
At the time Lincoln was speaking, those that might have considered themselves atheists or even agnostic were rare indeed.
But far from unknown amongst the elite intellectuals.
some of whom leant towards the Unitarian or deist beliefs.
Even those who despised formal religion, would likely not totally reject the existence of God.
America, and the world, were uniformly a God fearing people.
America was founded by, and for, those God fearing people.
A massive majority of Americans would have considered themselves to be some form of Christian.
They would have considered other religions as rare and curious or even heathen, with the possible exception Jews, who shared a God but not a saviour.
When they spoke about religion and God they meant Christianity.