You made a video showing it working. You say it's energy storage rather than friction that makes it jerk but, and this is what you are ignoring even though you actually made it happen, the vehicle moved to the right. At no point did it move left. It would keep moving right as long as you pulled the paper, jerking or not.
Your energy storage fetish is just covering up the fact that the model worked as predicted, as you proved.
I made the video exactly to show what happens for someone that can understand it.
It is clear both in video and in theory the front wheel (wheel on the right that is generator wheel) moves while the vehicle and back wheel are stationary meaning that energy from the front wheel is being stored.
You have a force and speed at the front wheel so you have power but the power input is not doing any work as it is being stored as elastic potential energy in the belt.
As the belt is stretched more and more storing energy the force at generator wheel with will be mirrored exactly at the motor wheel and at some point this force becomes so large that the wheel will slip allowing in that moment the back wheel to rotate forward powered by the energy stored in the belt and as that stored energy is being used up the acceleration slows down until the wheel sticks again and the cycle repeat.
A smaller energy storage capacity (stiffer belt) means the cycle will need to repeat much faster so fast that it will not be visible without a high speed camera (human brain will see that as continue motion same as 30 still photos in sequence per second look like smooth motion).
If wheel was not allowed to slip or reduce the slip to a level where power delivered by the energy storage can not cancel friction the vehicle will not move or move backwards so from right to left.
You just make the theoretical calculation of that simple system excluding energy storage and you see that vehicle has no chance to move forward unless of course like Naej you think more than 100% efficiency is possible.