Why are primary and supportive instruments in instrument flying different depending on the phase of flight?
Because during certain maneuvers, certain instruments provide more relevant information to a pilot than do others.

In straight and level flight, the heading indicator provides more relevant bank information as it tells you whether your current bank attitude is maintaining the desired heading.  The altimeter is providing the most relevant pitch information as it’s telling you whether your current pitch will maintain the correct altitude.  The airspeed indicator is the best indication for your required power as it’s telling you if the current power settings will maintain the desired airspeed.  Other instruments provide supporting information.  Here in straight and level flight the attitude indicator provides good supporting bank and pitch indications, the VSI provides supporting pitch information, etc.  Supporting instruments change to primary instruments in certain maneuvers and vice versa.

And, incidentally, in a established constant airspeed climb, your primary for bank is the heading indicator and your primary for pitch is the airspeed indicator.  Primary for power is either the tachometer, manifold pressure, torque, or fan speed gauges, depending on the aircraft you are flying.  Typically you would set to maximum continuous power in a light airplane for this situation.