For a simple sail it is impossible to extract power from the wind when the vehicle is moving at the speed of the wind or faster with the wind from behind. Surprisingly this is no longer true for the vehicle with the propeller and wheels to interact with the ground. This is the slightly confusing effect the whole discussion is about. For the discussion you can not start with the assumption that this is not possible - it is not a well known fact, but a more or less widespread misconception.
It is known that one can extract wind energy it the wind comes from a different direction. The wings of the prop move and the apparent wind is no longer coming from the front. So in principle they can extract wind power. So you see that with moving winds it is no longer clear one can not extract power. I would not explain the effect this way (the vehicle used a different principle), but it shows that a clever construction can do things a fixed sail can not do.
An ideal propeller is kind of working like a srew in a solid, though real world the efficiency is lower.
With such an ideallized prop the problem get transformes to something like 2 platforms / referenc surfaces moving relativ to each other. With a simple mechanical systen, like with wheels and pullies, it is relatively easy to understand that the difference in speed can be used to power a vehicle at any sensible speed and direction. This part may confuse a child, but is not really surprising.
It only take basic mechanics to understand. For understanding it may be easier to just calculate speeds and use the wheels, links and gears as conditions to link different movements. So all without, without looking at power or forces, just at position and coordinates. This usually leads to linear equations that are not so hard so solve. If such a mechanical system allows a movement (solution for the equations) it would do so with only friction forces when forced to by the boundary conditions (e.g. moving plantforms).
The arguments with motor power to decide which side is stronger tricky as motor power sometimes means maximum power and real motors are more limited in torque and not actual physical power (in W). This is especially true at low speed or starting from 0.
The point how good a prop can rally work is hard to calculate - I fully understant if one does not follow the math behind the limits there - that is hard core aero dynamics and may not have closed solutions. It is sensible that a suitable size prob can have enough efficiency to get at least a speed slightly higher than the wind speed. Working with a relatively low relative speed it should be even easier for a prop to work in water.
I looked at all explanations available in details but the current explanation is wrong.
Propeller can not magically have access to wind energy when wind speed is below vehicle speed and both are in the exact same direction.
A sail is the most efficient way to use wind energy and ideal case a sail that is at the same speed as the wind speed has nothing more to extract from the wind and you will call that a 100% efficient sail powered vehicle (not in real world just ideal case).
No matter what other device you use to replace the sail you will not be able to extract energy from the wind when vehicle and that device (propeller or anything else) drives above wind speed.
Think about it this way. Say you have a wind turbine on wheels and when stationary you extract the most you can from the wind (real wind turbine is just at most around 40% efficient but that is irrelevant). Now you start to move the wind turbine in the same direction as wind direction so when you get at half wind speed the wind turbine will experience just half of the wind speed so just 12.5% of the power level compared to stationery wind turbine.
When your turbine move at the same speed as wind it will experience zero wind speed so it will not be able to produce anything and above that things will not change as apparent wind will now be from the other side and if you try to access that by rotating the wind turbine 180 degree you will slow the vehicle down way more than you produce from the wind turbine.
The same applies here with a propeller used as a fan and the only difference is that pressure differential energy storage witch allows vehicle as you see in tests to exceed momentarily the wind speed may be a few minutes depending on design and amount of stored energy.
Yes a propeller works as a screw in a solid now imagine a piece of wood or butter
for lower friction where a screw stile propeller will advance and below this wood the vehicle moves on the ground. As long as wood moves above ground (floating) the vehicle can be powered by this wood but as soon as vehicle speed same as wood speed there is no longer any way to power the vehicle as taking power from vehicle wheel and transferring that to the screw will result in vehicle slowing down as not all power from the wheel can be transferred to the screw.
What you are confusing is the apparent wind speed direction relative to vehicle. Below wind speed vehicle will see a positive power available above wind speed the power will be negative meaning counter productive so vehicle will be slowed down rather than accelerated.
I know the limitations of a real motor but we are talking about power here so it is assumed the motor at whatever speed is capable of that power. The same is true about a generator so you need to remember that a generator powers the motor and while this is an electrical analogy using sprockets and a chain to transfer power from one wheel to the other will be constrained by the same rules.
Yes fluid dynamics is not easy (I had done both mechanics and fluid dynamic curses at university) but you do not need to know fluid dynamics (you will usually have a computer to simulate something like this) all you need to know is that if your only power source is the generator wheel that will not be able to provide the motor with more power than the breaking power.
Breaking (generating power) will be from the vehicle kinetic energy and if that kinetic energy is not put all back the vehicle will slow down.