The US government at all levels is controlled by two parties. While neither side is going to just hand over control to the other, they certainly aren't going to vote in a system that would allow a 3rd party to emerge.

There is no way such a system would ever be allowed to produce a ballot without a Republican and Democrat on it. If you're not on the ballot, you don't get elected. Full stop. There would be all sorts of bizarre measures and "tests" added to such a system before it was ever in place to make sure that only the existing parties would ever be on the ballot.