I have also traveled internationally with my child without the other parent. However that was prior to 9/11. Many parents on these forums have stated that they were unable to travel internationally without written permission (or a copy of a court order giving permission). So, I know it happens. One parent managed to get out of the US, but almost couldn't get on a plane BACK to the US. I think they were visiting Germany.
Anyway, one really has to assume that they need that written permission, because it DOES happen.
OK, but the point was that your absolute and unequivocal statement that, "without written permission[,] the airline still won't let her on the plane" is incorrect.