Because Runways Are Short (relatively speaking)
The simple answer comes down to the fact that you only have so much space to take off so getting off the ground as fast as possible is the most practical and most safe option.
Aircraft will fly when the wings generate sufficient lift to overcome the weight of the aircraft, this happens when there is enough airflow over the wing which in turns broadly happens with sufficient air speed.
Lets examine a counter case for a minute. Say you did a partial power takeoff, you only apply some throttle (enough to overcome drag) and start your roll slowly, you will be out of runway long before you reach your takeoff speed. That being said with sufficiently long runways you would be able to take off with less than full power.
Aircraft engines have no issue spooling up and running at full power for takeoff and climb out, as a matter of fact aircraft engines are generally better at running at high power settings for long periods of time since that is exactly what they are designed for.
Takeoffs also include a "decision speed" which should take full throttle breaking distance into account so you should be able to abort a takeoff if there is insufficient power or speed.
But, also, the POH says so...