I think that I may be making some progress here, but rather than this specific example, I’m trying to understand the analytical method that you are advocating. For me that is much more useful. I have made a list of the key points that I have picked up from what you have written. Am I on the right track here? If you do see any errors, could you correct the text as necessary.
1) Intuition plays a key role in classical mechanics.
I do not know if I will agree with that. There is not much if anything that was not studied in classical mechanics. Sometimes "intuition" seems to be wrong.
For example using kinematics only and not considering forces will not provide the correct answer.
To me it seems that people always imagine that the force is applied between vehicle body and input wheel not as it is the case between the input wheel and the ground.
The only time the force is applied between the vehicle body and output wheel is when stored energy is discharged in that case the vehicle is powered from inside and not form the exterior.
2) Before making any calculations, it is vital that the analyst chooses the correct frame of reference in which to perform those calculations. This can only be done through intuition, built up from years of experience. Inexperienced analysts can easily get this step wrong and choose the wrong frame of reference. In which case any conclusions that they draw from subsequent mathematical analysis will be invalid.
Again not sure we have the same definition for intuition.
Result will be the same independent of the reference frame chosen.
3) There are two classes of frames of reference, which are the accelerating and non-accelerating frames of reference.
For this particular example you can only have acceleration if either input or output wheels slip so unless that happens there will be no acceleration.
The applied force F1 needs to be large enough for slip to happen else if that is less nothing will happen other than belt will be stretched the input wheel will rotate correspondingly with the amount of stretch but vehicle body will not move relative to ground.
If output wheel (left one) slips first then vehicle will be accelerated to the left and F1 the applied force will be F1 = F2 + m*a so sum of 3'rd and 2'nd law.
If input wheel slips (right one) (this is what happens if you have the same type of wheels and and material since the input wheel already rotates and it will slip more easy) the you have F2 = F1 + m*a and F2 can be larger due to belt applying a force inside the vehicle between the vehicle body and output wheel and this force is possible because of stored energy.
As you see from this equations the gear ratio is not involved unless you look at the amount of stored energy as that will depend on the gear ratio.
4) In an accelerating frame of reference, objects within that frame obey Newton’s second law of motion.
As I mentioned 2'nd law is involved only if you allow wheel slip. And the direction the vehicle is accelerated depends on which wheel slips.
If the vehicle worked the way you think it works there will be no wheel slip but that is not the case in any experimental test where vehicle moves.
5) In a non-accelerating frame of reference, Newton’s second law is not relevant, because nothing is accelerating. In this case, Newton’s third law will apply to every object in that system, so every object in the system will be subject to a net zero balance of forces. In the case of two forces acting on an object they must necessarily be equal and opposite. This is what Newton meant by stating "To every action, there is always opposed an equal reaction”.
That is not quite true. It is true that net forces for this particular problem is zero in an non accelerating reference frame.
But if you have a proper (working) gearbox that means 3 points of contact so vehicle body connected to ground. Then the applied F1 will be between vehicle body and input wheel and the F2 will be between vehicle body and output wheel and at 2:1 gear ratio the F2 = 2*F1 and that will be the case in an non accelerating frame of reference.
If not convinced by this think about a balance with fulcrum the third point offset as in image below.
Nothing will need to move for the input force to be smaller than output force and Newton's 3'rd law works here as two separate loops with F1 input force having a pair at the fulcrum and the other pair F2 output also relative to fulcrum.
But example a has no 3'rd point the fulcrum equivalent so with that missing input and output force can only be equal.
6) Some mechanical systems can alternate between being in accelerating and non-accelerating frames of reference. The analyst needs to take great care here to allocate the correct frame of reference to each moment in time. Inexperienced and even many experienced analysts can get this wrong, and is one of the most difficult skills to master.
Yes that is the case here when vehicle moves to the right against the applied force as the vehicle will accelerate in burst many times per second so fast that it looks like smooth motion to the slow human brain.
Maybe it is a difficult skill but as soon as you see a device moving in the opposite direction of applied force you will know that some other energy source is present like in this case a small energy storage device.
Because you know from experience and Newton's 3'rd law that for every action there is an equal and opposite reaction so you can not have an unpowered item like this vehicle pushing with higher force F2 against the applied F1 as that will also violate the energy conservation.
The energy conservation is not violated in this case only because energy is stored so treadmill moves say 1cm while vehicle is not moving at all but this 1cm and the F1 was stored as elastic energy in the rubber band then in the next moment this stored energy could be converted in kinetic energy in the opposite direction and if there was no friction with just this initial energy the vehicle can maintain forever the gained kinetic energy but since there is friction the cycle of charge discharge needs to repeat all the time to maintain an average speed. Speed of this vehicle is not constant but it is not possible for humans to detect that without faster more sensitive measurement equipment.