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Only the first two Presidential elections were free of party, and that was due to the stature of Washington. You are wrong about the next two. They were rip roaring political fights between the Federalists and the Republicans. The 4th one in 1800 is the gold standard for no holds barred politics. They changed after the fourth because of Aaron Burr going for the top job in the House even thou he was running as Jefferson's VP. They did not want to repeat this.
They fought hard against each other because they believed the other side was out to annihilate theirs. The Federalists believed that Jefferson would roll out the guillotines if he won. The Republicans truly believed Hamilton or Adams would crown himself king if he won.
A good example for today, is Alexander Hamilton in 1800. Thou Jefferson was his arch foe, he supported him over Burr because he believed Jefferson loved his country and Burr only loved himself. The next four years showed how correct Hamilton was in his assessment.