Are we reading the same constitution?

Article II, Section 1

> The Person having the greatest Number of Votes shall be the President

A party is a thing where multiple elected officials band together in a persistent coalition. The section you're quoting from only applies to a single elected office in the whole country. Are only two parties are going to run candidates for President when there are five or more parties in the legislature?

On top of that, that section applies to how the votes of the electoral college delegates are counted. It doesn't specify how the electoral college delegates are chosen, which it leaves up to the states. There are plenty of interesting ways of choosing them that don't result in a structural incentive for a two-party race.

I'll tell Hillary Clinton, she'll be thrilled.