Assuming the Newtonian framework F=dp/dt, p=mv, dW=Fdx, as well as constant mass, then Fdx=dpdx/dt=mvdv and integrating both sides gives deltaW=1/2mvf^2-1/2mvi^2+constant. So the amount of work to move the object from x1 to x2 is proportional to the difference of the square of the initial to final velocity squared up to a constant. This we define to be the change in kinetic energy.
But as others have mentioned this is only as intuitive as F=ma, or p=mv.
In my view, at least classically it's just a matter of definitions then. If our definitions of energy differ, the only thing we will experimentally agree on is the equation of motion, and even then up to a frame transformation.