This debate has been ongoing for decades, and only got more intense with the airport police and LAPD having a turf war at the airport over command and incident reporting structure since the El Al ticket counter shooting and 9/11.
Sure the LAPD could cover the airport, but it would need to absorb a bunch of bodies. LAX Airport police today has 430 sworn officers at the moment and still is hiring, so its not a small operation by any means.