Article II, Section 1
> The Person having the greatest Number of Votes shall be the President
Article II, Section 1
> The Person having the greatest Number of Votes shall be the President
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 don't think it's a coincidence that every US state is structured as a smaller mirror of the federal government.
Using score voting instead of FPTP for state-level offices would be a straightforward legislative change in many states and still not require any change to the US Constitution even in the states where it would require a change to the state constitution, which is generally a much lower bar to overcome than a federal constitutional amendment.