The State Department is mischaracterizing the law, which requires only that the US citizen "bear" a valid US passport, not that the US citizen "use" the passport.
This law, 8 USC 1185(b), used to have a fairly stiff penalty, and it used to apply only in time of war. When the wartime element was removed in 1978, so was the penalty. It now reads
Except as otherwise provided by the President and subject to such limitations and exceptions as the President may authorize and prescribe, it shall be unlawful for any citizen of the United States to depart from or enter, or attempt to depart from or enter, the United States unless he bears a valid United States passport.
If a US citizen attempts to leave the US without a valid US passport, there is a very small chance that the departure could be prevented by a CBP officer, but in the normal course of affairs the traveler would not even encounter a CBP officer, so the possibility is very remote indeed.
There is nothing, however, that prevents a US citizen who also holds a passport issued by another country from using the other passport while also carrying a valid US passport.