Wrong. DHS tells the Airlines your ESTA Status, so they won't let you board if you don't have a valid one. And if you don't have one, the Airlines will check for a Visa
"DHS communicates a traveler’s ESTA status to the carriers. However, DHS recommends that travelers print out the ESTA application response as a record of their ESTA application number to confirm their ESTA status."
They also have DHS Agents on the departure Airports, which already tell the Airlines which Passengers aren't allowed to board. If the Airlines violate against this, they face severe consequences like a ban from US Airspace
