True.. adding flaps produces more drag. But the increase in lift is important here and can't simply be discounted. The additional lift that is created with flaps extended can be directed opposite the yaw induced by the engine failure. This is accomplished by banking into the operative engine to change the lift vector. This leaves more rudder available.

It may be that the induced drag caused by creating more lift would be too much for the aircraft to overcome. Its important here to empasize the difference between aircraft performance and aircraft control. The airplane may not be able to climb... so it slowly crashes in to the ground.. straight ahead and under directional control.