Why did the two party system nearly disappear by 1816?
The time that you are asking about is typically called the "Era of Good Feelings." It came about largely for two reasons. First, the impact of "winning" the War of 1812 cannot be understated. Americans felt good about themselves and felt united as a people having just stood up to a world power. This helped to reduce partisanship.
Second, the Federalist Party had essentially committed suicide during the War of 1812. It had held the Hartford Convention in which it talked about seceding and seemed to take the British side in the war. It did this largely because it was a Northeastern party that depended on trade with Britain. The fact that the Federalists seemed to have been disloyal led to their death as a party.
For these two reasons, there was little partisan competition by 1816.