The answer is quite simple and has very little to do with lift. The useful power from a thrust $F_t$ on a body with speed $v$ is given by $P=F_tv\cos(\theta)$, where $\theta$ is the angle between the direction of thrust and the direction of movement. This is equal to the difference between the angle of attack and the chord angle.
In other words, for a given thrust from the on the propeller, you get the most energy out of it when the propeller is pointed directly into the relative wind (i.e. for a specific angle of attack). The farther apart these directions are, the more power is being wasted. An extreme example of this is slow flight, where full power may still be inadequate to hold level flight at extreme angles of attack.
In other words, while climbing in a level attitude is possible, it is inefficient in the same way that slow flight is, though to a lesser extent. You get more performance with the same power by allowing the nose to pitch up.
The wings also have a most-efficient angle of attack (best lift-to-drag ratio), which further enforces the idea that certain angles of attack are more efficient than others.