This method allows for the best kind of voting, one with effects and one that expresses people's opinions. A voter could vote his conscience, which could be a third-party, _and_ ensure that his honest vote did not help his least-favorite candidate, since he would also vote for the two-party candidate he liked the best. Additionally, we would be able to take an accurate barometer of a particular candidate's popularity, which is especially important in determining eligibility for federal matching funds.
If, after implementing this system, the two parties remain in power...well, that means that the American people _want_ these two somewhat-similar groups the way they are. However, we might surprise ourselves and give a third-party candidate more approval votes than either major party candidate.