How does the power distribution change for a turbofan engine from being stationary to going at full speed? I have found out that a turbofan engine of say an F-16 is producing zero power when stationary since $Power = Thrust × Velocity$. But assuming that the engine is producing the same thrust, it appears that regardless of whether the F-16 is stationary or moving, the engine is still consuming the same amount of fuel and therefore there is power output that is constant in both cases. I also found that the power I tried to calculate from the equation above is the propulsive power, so there must be something else that is consuming the rest of the power.

To make this simple, I will use some numbers. Let's assume a turbofan engine is capable of producing 100 kN of maximum thrust with no afterburner while consuming  4 kg of fuel per second. For the sake of simplicity I'm gonna use a figure of 30% thermal to mechanical efficiency for the engine that is constant through all of the following cases and which means that  there is 30% $×$ 4 kg $×$ 42  MJ/kg = $50 MW$ of mechanical power available at all times. With 43 MJ/kg being the heat value of the jet fuel.

Case 1: The engine is stationary and producing max thrust and therefore the propulsive power is zero so I assume that the 50 MW available is completely used in heating and accelerating the exhaust.

Case 2: The aircraft is accelerating and is now at half its max speed at sea level which is say 200 $m/s$. The propulsive power is now = 100,000 kN × 200 = $20$ $MW$ and so I'm also assuming that the other 30 MW remaining of the 50 MW available are still being consumed in accelerating and heating the exhaust.

Case 3: The aircraft reached its max speed at 400 $m/s$ at sea level. The propulsive power is now = 100,000 kN × 400 = $40$ $MW$ and as before, 10 MW are consumed by the exhaust.

So, if I understand this correctly, the power distribution goes from 0% propulsive power and 100% exhaust power when the engine is stionary to 40% propulsive power and 60% exhaust power when the aircraft is at half full speed and continues to shift the power available to propulsive power at higher speed until it is 80% propulsive power and 20% exhaust power. My question now is: In simple terms and with acceptable approximations, does this shift that redistributes the power from exhaust to the propulsive power as the speed increases really happen?
Your definitions are correct and your derivations correctly use the law of conservation of energy, so yes, the power distribution change does happen.

The more important question is what does this actually mean. Energy is a rather curious quantity in that while it is conserved in each reference frame (including non-inertial as long as appropriate potential is included), the values of kinetic and potential energy will be different in each!

You can look at all the cases in the reference frame of the aircraft and then the propulsive power will always be zero, because the velocity is—by definition of the reference frame—zero. But the very same situation viewed from the reference frame of the Earth it will be as you describe, and from the reference frame of the air mass it will be similar, but the values will differ slightly as the reference frame move relative to each other at the speed of the wind.

Note that enthalpy does not vary in coordinate transformation, so the sum of the power towards moving the aircraft and towards accelerating the air stream is will be the same in all the reference frames. Most easily calculated in the reference frame of the aircraft from difference in kinetic energy between the intake and the exhaust. This is the output power of the engine itself, and comparing this with the heating value of the fuel will give you the thermodynamic efficiency. Thermodynamic efficiency of turbine engines increases with flight speed due to pressure recovery, i.e. the ram air pressure increasing the effective compression ratio.

Also note that this variance in propulsive power is there for all kinds of propulsion systems, including propellers and wheels of ground vehicles. The propulsive power is always zero at standstill, and how quickly it increases depends on the reactive mass they have available. The more reactive mass, the larger the propulsive efficiency (= propulsive power to output power) will be, and the faster it will increase with velocity. Propellers have some more reaction mass available due to larger diameter, and wheels have whole Earth available, but their thrust is still limited by the surface friction and the maximum torque of the engine (and the friction of the clutch for engines that don't produce torque at zero RPM).