Why are ambient pressure and burner can pressure always the inputs on a hydromechanical fuel controlling system?
Hydromechanical Fuel Control Units (FCUs) typically do use compressor discharge air pressure as an input.

As I recall, the FCUs that I am familiar with have had at least the following inputs in addition to the control lever commands:


Compressor Rotation Speed
Compressor Discharge Air Pressure
Ambient Air Pressure


As both a mechanic and pilot I have studied many FCUs, and I do believe the inputs listed above are standard across all those with which I am familiar. This is certainly true of the types that I have studied most recently and the systems which I currently fly. Note that while these FCUs may be older designs—in some cases—they are commonly used on new production turbines.

The following are excerpts from some of my training manuals concerning a Bendix FCU:




  The pneumatic section of the FCU determines
  the flow rate of fuel to the engine for all operations.
  The power levers control engine
  power from idle through takeoff power by operation
  of the gas generator (N1) governor in
  the FCU. Increasing N1 rpm results in increased
  engine power.
  
  For explanation purposes, consider the N1 governor
  bellows as a diaphragm. P3 air is introduced
  into the bellows in a manner that sets up
  a differential pressure on each side of the diaphragm.
  Therefore, any change in P3 pressure
  will move the diaphragm. When pressure
  is increased, the fuel-metering valve attached
  to the bellows will move in an opening direction
  to increase fuel flow and increase N1 rpm.
  
  As P3 pressure decreases, fuel flow also decreases
  which reduces the N1 rpm. The N1
  governor increases or decreases P3 pressure in
  the bellows by varying the opening of relief
  orifices in the bellows.
  
  The FCU controls engine power by maintaining
  the requested N1 rpm through the N1 governor.
  If actual N1 rpm is lower than the desired
  setting, the N1 governor closes the P3 orifice,
  allowing pressure to increase. As the pressure
  increases, the diaphragm moves to open the
  metering valve, increasing fuel flow, which in
  turn increases N1 rpm to the speed requested by
  the governor. When N1 rpm reaches the desired
  speed, the governor adjusts the P3 orifice to reduce
  pneumatic pressure to match the fuel pressure
  required to maintain the desired N1 rpm.