You cannot apply a force at F1 in case (a), because the wheel will turn, and by yielding reduce the force to zero.
The wheel will not turn unless the belt is elastic and the stretching of the belt allows the wheel to rotate while force will increase.
You are imagining that the axle friction is so high and that is the reason why the belt stretched so much in my experiment but that is not the reason (not even close).
The F1 wants to push the vehicle to the left while the left wheel wants to push the vehicle to the right with the force F2 equal and opposite to F1.
It is the same as case (b)
I think you are imagining that F1 only wants to turn the wheel and not push the vehicle in that same direction and that will be true if the belt was not connected and the wheel axle had no friction.
But due to the way the belt is connected it is basically a locked gearbox no different from having the wheels welded to the body.
The first case is not floating as it has a wheel resting on the solid red block, and the other wheel is resting on the belt, therefore not floating either.
The wheels can rotate independently of the body but not independent of each other due to belt.
The body drawn in blue is not connected to anything. If I were to connect the body to ground right in between the red box and the treadmill then F2 can be 2xF1 (assuming 2:1 gear ratio) and so that back wheel can push the red box with a force twice as large relative to grout as the input F1 also relative to ground.
Same thing for case (b) where the cylinder body will need to be connected to ground.
You can not do force multiplication with just two points of contact and you need a minimum of 3 points in order to do that no matter the device.
Just have a search on google images for "torque multiplier wrench" and you will see that all of them require 3 points of contact in order to work.