Most of what the rocket has to increase to take an object from the earth's surface to LEO. If you calculate the potential energy of a rocket in LEO and compare it to its kinetic energy, you'll find the latter is much larger. (I've already done this and posted it here.)

Something with exactly escape velocity from a body always has zero total energy. That's the definition of escape velocity.

How so?

A rocket at infinite distance from the Earth will have infinite PE and can have zero KE.