Maybe we can.... I don't really see the problem. I guess you're trying to MAKE it some sort of "requirement," but the premise in the original post is fallacious at best.
Lincoln had zero time in the US senate, and not much high-profile political experience in Washington, overall. He's still ranked as one of the greatest presidents and that is on the strength of being an intelligent lawyer with forward-thinking decisions, a conviction to unite people, and his tool of diplomatic eloquence.
There are a good number of other presidents that don't have the qualifications that you seem to be looking for. Eisenhower .... G. W. Bush.... Woodrow Wilson... McKinley ... etc etc.