Radius of Circular orbit of a Satellite around Earth always constant (if Space was a perfect vacuum) WHY?

You are correct that the Earth is constantly pulling orbiting satellites toward it. But because of their lateral velocity, they are able to 'miss' hitting into the Earth.

See the section "The 'weightless astronaut' paradox in this link. I also like the quote at the beginning as it turns your question into a statement of fact.

Being in orbit is like being infatuated – you are constantly falling, but you aren’t getting closer.

Forget about the Earth-Satellite system, your question is much simpler: you are asking about circular motion in general.

It is a basic notion that, for a uniform circular motion, a centripetal acceleration is required. This acceleration changes the direction of the velocity, but not its modulus.

Because acceleration is the change in velocity, but vectors!

$$vec{a}=frac{Deltavec{v}}{Delta t}$$

A vector can change if the modulus changes, but also if the direction changes.

A perpendicular acceleration does not change the speed, it only changes direction of motion. This is a basic thing you must learn before you study any dyunamics. Kinematics are more fundamental.