This calculation neglegts the force on the turbine. This gives an additional energy as drag force times vehichle speed.
Even for just a passive sail the energy is vehichle speed times force and thus
0.5 * air density * area * (wind speed - vehicle speed)^2 * vehicle speed
The passive sail is however not the best that can be done. So the theoretical available energy can be larger than that. The blackbird vehicle showed one way to do that. A sail boat going at a suitable angle is another way - though it complicates the area part. Anyway it still gets power when the speed component parallel to the wind is faster than the wind speed so there is no inherent zero at that speed. Yes it is sideways, but for a though expoeriment one can consider the sideways area as still part of the vehicle size, so the movement would be only inside (like the fan going in circles).
One could consider the fan blades as sails going zig-zag. So this kind of circumvents the going straight in the direction of the wind.
The turbine is supplied by wind power so you can not add that again.
Everything on a wind only powered vehicle is supplied by the wind power.
You can search research papers or similar and see that any direct down wind sail vehicle powered only by the wind will use this wind power formula
0.5 * air density * area * (wind speed - vehicle speed)^3For a wind turbine the same formula is used except you do not need that vehicle speed subtracted as the wind turbine is fixed to the ground and this is the ideal wind power so to this you will add turbine and generator efficiency.
I designed my own turbine so I worked with this sort of formulas before.
The passive sail is actually the most efficient and an ideal one will be 100% efficient.
That is why no vehicle exceeding wind speed directly downwind (special case) powered only by the wind can not exceed wind speed.
The way blackbird is doing that is by storing energy in pressure differential while it is below wind speed and then use that stored energy to be able to accelerate well past wind speed.
Sail boat going at an angle is a geometric problem as the sail will still have access to wind speed since it will not be driving faster than wind relative to wind direction. Best case is driving perpendicular to wind direction as then you have always access to same wind power as vehicle is not moving at all in the wind direction.
And while driving at an angle the vehicle kinetic energy is what will store energy and then in an ideal case no friction you can just change direction to direct down wind and maintain that higher wind speed indefinitely (no losses) in real vehicle they will slow down same as blackbird will also slow down.
Since kinetic energy is directly related to vehicle speed as soon as a real sail vehicle changes direction to direct down wind it will start to slow down but on the blackbird energy is stored in a separate energy storage device the pressure differential.
If you have some electrical knowledge the equivalent of kinetic energy is a capacitor and the analog to pressure differential is an inductor.