I can deal with the aircraft exerting the same force on the air as the air exerts on the aircraft, but is the force of the aircraft on the air not taken up by viscosity and eventually heating?
No. Because momentum is conserved.
If you accelerate, say, 1 kg of air to 1000 m/s every second, you are exerting a force of 1000 newtons - and spending 500 kW energy to do so.
If that 1 kg of air then exerts force on and entrains 9 kg air then momentum is conserved. 10 kg air per second will be moving at 100 m/s. And it carries only 50 kW of energy.
The other 450 kW energy was lost to viscous heating of the air. But the force and momentum is still there in its entirety.
As the air goes on moving, its energy decreases as increasing amount of air is made to move at increasingly low speeds. But the momentum remains there until it is transferred to ground.