Why do so few people seem to realize that "escape velocity" is relevant only for unpowered objects and vehicles?

you have a point. With an infinite amount of power, a spaceship could boost at 100 MPH straight up and escape the planet easily. 

However, at the point where the ship shuts off it's power, it has to have escape velocity for it's altitude or it will fall back. 

But I disagree with the "so few people" part....

You seem to be mistaken

Unpowered Vehicles have a propensity to not move

The Escape Velocity for Earthbound Vehicles is 17, 500 mph

That would get the Vessel into Orbit

You would be doing well to throw something that hard !!

For Trans Lunar Injection it is 25, 000 mph

For crossing the Solar System, the Minimum is 35, 000 mph

The clue is in the word Velocity

Because that isn't true. Why do you think it is?

most people cant make change or ten dollars let alone know what escape velocity is ..

I heard from one guy that the moon landing couldnt happen because they couldnt bring enough fuel they think that in space in order to go forward you need to press the gas pedal..

Most people don't even have any idea what escape velocity means.

You are right. A powered rocket can leave earth's gravity going at any speed. Nasa chooses a flight path that uses the least fuel.

Because most people don't have a clue or an interest in Classical Mechanics. Most people think rockets go straight up and turn left to get into an orbit. And they have no clue why they go into orbit either but they do know going into orbit is what rockets ultimately do.

Escape velocity refers specifically to unpowered objects. It has no relevance to powered machinery.

If a machine such as the Dawn probe wanted to use an ion thruster for months at a time, it could power it's way out of the solar system using continuous thrust. But this is not the same concept as using a traditional chemical thruster to give a payload enough thrust to reach a velocity at which it could coast right out of the solar system without any further thrust.

you have a point. With an infinite amount of power, a spaceship could boost at 100 MPH straight up and escape the planet easily.

However, at the point where the ship shuts off it's power, it has to have escape velocity for it's altitude or it will fall back.

But I disagree with the "so few people" part....

Escape velocity is relevant to powered vehicles, to unpowered vehicles, to objects thrown out by accidents, etc.

Escape velocity does not care.

As you move away from a centre of gravity, the escape velocity relative to that object gets lower. Therefore, it is possible to power yourself at slow speed to a greater distance (where escape speed is lower), and then accelerate to that escape speed. You must still reach that escape speed if you want to escape.