tl;dr: Two intuitive arguments derive KE ∝ v² without invoking work or mgh. The first uses a spring pushing two equal-mass boxes apart, combined with conservation of momentum and Galilean invariance of potential energy, to show that doubling velocity quadruples kinetic energy. The second uses a constant gravitational field and energy conservation: comparing dropping an object in four stages versus one, and launching in reverse, forces KE(v) = 4·KE(v/2).