Why isn't the airspeed indicator the primary instrument for power during a constant speed climb?
Airspeed indicator is a primary instrument for pitch in that row, so it can't be also primary for power.

The point of constant speed climb is that you select power and maintain speed with pitch, getting whatever climb rate that power can support. Most often the point is selecting "climb" or "maximum continuous" power and climbing as fast as possible without overstressing the engine.

The only difference with variable pitch prop is that it decouples rpm from engine power, so you can't use rpm and are left with manifold pressure only (for piston; I am not sure what the correct reference is for turbine).

Note that two lines lower there is a constant rate climb and there airspeed is primary reference for power, because there you adjust pitch to maintain desired vertical speed and power to maintain airspeed. But since you shouldn't use more power than the recommended climb setting anyway, the constant speed climb gives you better performance.