Normally going out of LAX, I would say united. But based on your travel:
LAX - London (AA or BA have non-stops)
LAX - Maui (AA, United, and Hawaiian Airlines)
LAX - NY (Delta and AA have more non-stops to NY then United)
LAX - Aruba (don't know this one to be honest)
I would say the winner based on this situation would be AA only because you have more options for London and NY on AA from LAX. YMMV