So what if the United States was organized along Roman Republican lines, with the Senate controling most of the day to day activities of government, with the president only being elected by the senate durring times of war, or national crisis, elected by the senate from the United States Army and serving as "supreme comander of the armed forces".
Notable OTL presidents who might make good candidates for this role:
Zachary Taylor
Dwight D. Eisenhower
Andrew Jackson