To me, the fact that a normal plane climbs, at a relatively constant speed, when power is increased, is simply the consequence of the pitch stability mechanism which every certified plane has.
The elevator has what is called decalage which AIUI means that the turning moment due to the change of elevator AoA about the pitch axis has to exceed the turning moment due to the change of wing AoA about the pitch axis. If you didn't have this pitch stability (which is in effect a negative feedback control system where airspeed is the setpoint) then the plane would be virtually uncontrollable manually.
I think that the cases where you are flying at say 65kt, trimmed, at low power, and increasing power causes the plane to nearly stall, are because one is flying on the back of the curve at 65kt (on that particular type). Otherwise, can a plane with such a large moment (due to power changes) about its pitch axis get certified?