“Although… the Chief Magistrate must almost of necessity be chosen by a party and stand pledged to its principles and measures, yet in his official action he should not be the President of a party only, but of the whole people of the United States.”
The President (and VP) is the only federal official that is elected by a nation-wide constituency. As such, he is the only one that understands plainly that he must govern for all of the people. This is why good presidents will at times “break from their party” and make deals with the other side, sometimes angering their base. But if those presidents are confident that their actions are in the best interest of their unique constituency, they can weather the storm of revolt in thier own party. After all, the party will usually come around and welcome their president with open arms. If not, who cares? At the end of the day, the president is the leader of his party.