I don't know where is the doubt about the airline checking what your status was in the country. If someone doubts it and has dual citizenship, please try to board the plane with the foreign passport and see if they let you in without showing what your status was in the US. Even if the foreign passport lets you go to the destination.
As a test, I used my foreign passport last week when leaving the US (online airline checking process). I was never asked what my status is in the US by the airline. As I suspected, it was Canadian immigration that asked what my status was in US, to which I simply answered "I'am a citizen". On my return I used my US passport for first time with no hassles whatsover. CBP officer was friendly and even commented on how the quality of some food is higher in Canada (I had declared some cheeses I had bought in Canada).