Klinkenhofen,
steady state effectively means "no acceleration".
For an aircraft to climb in a steady state, the forces striving to accelerate the aircraft upward have to equal the forces striving to accelerate the aircraft downwards. Force balance means no acceleration means steady state.
If the upward forces were in fact greater than the downward forces, as you suggest, the aircraft would be accelerating upwards and not in a steady state climb at all.