Do we need Escape Velocity [duplicate]

This may seem silly a lot but I really need some clarification on the necessity of escape velocity for a rocket leaving the Earth's gravity. A stone thrown vertically upwards reaches to a certain height before falling to the ground. The height at which the total potential energy gained by the stone is equal to the sum of the kinetic energy at the ground and the work done on it during launch. Since, after throwing it upwards, no external work is added, the stone WILL return to the ground in accordance with the law of energy conservation (velocity < escape velocity). But in a rocket, continuous work is being added from its booster motors. Then why is it necessary for it to acquire escape velocity in order to leave the Earth? What will make it stop from leaving? Consider the following hypothetical case:

A hypothetical situation:

Suppose we build a long vertical shaft on the Earth's surface, reaching into deep space and a person starts climbing on it (just like Jack on a bean stalk). Will he also require escape velocity to leave the Earth's gravity? If no, then how is it any different from a conventional rocket launch?