Dear Colin and Nam,
This question illustrates the conservative nature of the gravitational field. Because the track is frictionless, the only external force on the cart is gravity, so all the track is doing is controlling the path taken by the cart in the field, and because the field is conservative, the change in kinetic energy of the cart has to equal the product of the magnitude of the field and the vertical displacement of the cart, no matter what shape the track takes to get there. Conservation of energy is the correct principle to apply because a conservative force is applied to the cart.
I agree with Nam that the scalar kinematic equation has a different status from the other vector kinematic equations. I think it is perhaps overlooked that the product of acceleration and displacement in the equation is actually a vector dot product, corresponding to the twice the work done on the particle by the average net external force experienced by the particle as it moves from its initial to final positions divided by the mass of the particle. I would argue that using this kinematic equation implies conservation of energy, work done by a net force on a free particle changes its kinetic energy.