This is sort of right, but I need to niggle a bit, because there's a lot of errors in it.
First of all, angle of attack is not the same as down pitch. The angle of attack is the angle between the chord line of the wing and the incoming airflow. The pitch angle is between the longitudinal axis of the plane and the horizontal. Increasing the angle of attack does increase your lift, but it means pulling back on the stick, and it does not increase your speed in relation to the ground.
What increasing speed does is indeed increase Lift for a given angle of attack, wich is why you need to decrease the angle of attack to maintain your flightpath angle while increasing speed, but that's not the main reason you need to push forward.
Longitudinally stable aircraft, which almost all civilian aircraft are, are designed so that the increased lift creates a pitching moment that wants to pitch up the aircraft and return it to a lower airspeed. So not only do you need to push forward to change your angle of attack, you need to keep pushing forward (or trim the aircraft) to keep flying straight at an increased airspeed. This also means that in an accelerating descent or dive, you need to push forward ever harder, which makes it difficult to do.