Then how will the airlines know you will be admitted at your destination?
There is no such thing as exit immigration in the US.
Simple. The airline will know the same way that they will know for any other passenger. I will submit the appropriate document that will allow admittance.
If my U.S. Passport contains the appropriate visas, that's it. If not, THEN I would show the other passport that allows admittance. Two seperate issues - US passport presented so I can legally leave the U.S., other passport to prove I can be admitted at the other end. Simple.