Other people call it thrust - and you call it a pressure differential - but the question is on the energy storage.
Rather than running around with words - where we all seem to get nowhere - it would make life so much easier if you could provide the formula which tells us how much energy is stored.
If this was a non compressible fluid like water then there will be no pressure differential and no energy storage just thrust.
Here you are exactly wrong. "non compressible" is just a term, not reality. Just like everything in existence water can be compressed. It just doesn't change volume much when doing so. This is how sound waves travel in water, by compression. So there will be compression of the water behind a propeller and therefore energy storage. Not that it matters one whit. The tiny energy stored in the air or water around a propeller is insignificant compared to the energy needed to propel the vehicle.
Simplest analogy that I can think of right now and it is visual enough will be pushing a vehicle with another vehicle having just a solid bar connecting them (non compressible fluid like water) or having a spring between them (compressible fluid like air).
Just like water solids are compressible and will store energy.
The spring is an energy storage device and it is moving together with the pushed vehicle. As any analogy it has limitations but a compressible fluid can be used to store energy and a propeller can be used to increase the pressure differential thus energy can be stored.
So are you saying that if the propeller were in water and the wheels were on land, the vehicle would exceed the speed of the water easily with no storage?
This may be hard for some to understand as most will think that there is no way to maintain this pressure differential and there is not much I can do about this.
No, it is you who is saying the pressure can't be maintained. We all say the pressure storage is irrelevant and only exists as a result of the propeller pushing against the wind. All of your equations are wrong because they don't take into account the speed of the air pushed through the propeller.
The wind does push against the prop, it pushes the air which is exiting the prop. The net speed of the air pushing on the prop is the sum of the wind speed and the air speed from the prop relative to the prop. The net force on the propeller then is set by the sum of the wind speed minus the speed of the vehicle PLUS the speed of the air through the propeller relative to the vehicle.
The main point I want to make since this is ester to understand (at least is what I was thinking) is that there is no wind power available to a direct down wind traveling at or above wind speed.
Unless the vehicle is pushing against the wind.
I provided the correct equation and anyone that is not agreeing with that equation is welcome to provide the correct one.
You provided a correct equation for the force from the wind (except you didn't consider the speed of the propeller exhaust). You don't understand at all how much power is exerted by the wheels of the vehicle and won't even discuss that. You keep talking about the "available power in the wind" which is not the question.
Nobody can claim to understand a wind powered only vehicle without being able to provide an equation describing the amount of wind power available to vehicle.
That's not the issue being discussed by anyone but you. The power generated by the wheels is zero if the vehicle is not moving wrt the ground.
In addition you did not consider the exhaust speed in your wind power equation. So when the propeller is pushing against the wind this provides more available power than you are calculating!