> This is completely wrong.
On the contrary, it's completely right.
> Escape velocity is the speed, which is required to break free of the gravitational pull, BUT in the case that no additional propulsion is available.
You just changed the subject, and we could always branch out into a discussion of perpetual acceleration sources, but that would depart the realm of reality. The OP's question asked about reality.
> You can leave the earth's gravitational pull with ANY speed.
As a matter of fact, no, that's not possible -- unless you have a continuous energy source and don't care how much energy you expend. Which is your new topic, not the topic of current propulsion methods, the topic of the OP's post.
Without benefit of something exotic like a space elevator you can stay in place, stationary, 100 kilometers above the earth's surface, motionless with respect to the surface, in perpetuity, but only by a constant expenditure of energy. But that's a different topic.
> No, speed is irrelevant.
Without invoking technologies that don't exist, speed is completely relevant.
> What is important is to have a constant force ...
Be careful that you don't confuse static forces and those created by a dynamic source of acceleration that require an expenditure of energy. Static forces require no energy, but most accelerations involving motion do (apart from the class of friction-free orbits).
My reply answered the OP's inquiry, not the science-fiction idea that you could have an infinite, inexhaustible energy source.