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Your wrong. Of course the U.S. recognizes dual citizenship.

The modern day State Department rules discusses the problems associated with dual citizenship:

7 FAM 081: U.S. Policy on Dual Nationality:
(e)While recognizing the existence of dual nationality, the U.S. Government does not encourage it as a matter of policy because of the problems it may cause. Dual nationality may hamper efforts by the U.S. Government to provide diplomatic and consular protection to individuals overseas. When a U.S. citizen is in the other country of their dual nationality, that country has a predominant claim on the person.
...
the U.S. Supreme Court has stated that dual nationality is a "status long recognized in the law" and that "a person may have and exercise rights of nationality in two countries and be subject to the responsibilities of both." See Kawakita v. United States, 343 U.S. 717 (1952).

http://www.state.gov/documents/organization/86563.pdf

You're talking about a state department advisory/policy, not a law passed by Congress. The state department of course understands the concept, and advises U.S. citizens as to what it can mean when they travel abroad. But there is no U.S. law defining it or addressing what it is or what it means in any manner. This is a bureaucrats rule that has no force of law whatever.

Which takes us right back to my point that there is no U.S. law addressing or even defining what constitutes dual citizenship. Even the state department only says that you're a dual citizen if another country claims you as a citizen. But there is no test, under American law, for when a U.S. citizen can be considered a citizen of another country. Or rather, there is nothing in U.S. law that would invalidate a claim of citizenship from another country over a U.S. citizen.

Until we define that, a law that disqualifies a candidate because of "dual-citizenship" is completely open to gaming by the other nations. Cripe, use your head here. I'm not disagreeing with the concept of what you want to do. I'm trying to tell you what must be done to effectuate it.

Personally, as a starting point, I think it should be a violation of U.S. law for someone to hold a U.S. passport while also having the passport of another country.