why is angular velocity perpendicular to the plane of rotation ?

Angular velocity is a vector quantity. Any vector quantity has both magnitude and direction. The direction of angular velocity follows the "Right Hand Screw Rule", i.e. as you said, the direction vector is perpendicular to the plane of rotation.

Remember that, in general, a change of a vector quantity can be either a change of magnitude or a change of direction of the vector, or changes of both. This applies to angular velocity too.

Just consider a case where a disk is rotating counterclockwise in a horizontal plane. Using the "Right Hand Screw Rule", the direction is then pointing vertically upward. If a torque is applied to the disk and makes it (the plane of rotation) inclined at an angle, say 10 degrees, to the horizontal, but without any change of rotational speed, the direction vector thus moves from "vertical" to "10 degrees from the vertical". There is a "change of angular velocity" ( a change of direction without a change of magnitude). Since angular velocity is related to angular momentum, a change of angular velocity leads to a change of angular momentum,

Should you use your "clockwise and counterclockwise" concept, there would NOT be any change in angular velocity, as the disk is still rotating counterclockwise with the same angular speed. Then the application of a torque resulted in no effect on the rotating disk (or say it conversely, no torque is needed to change the plane of rotation of the disk). This is surely against our common physical experience. A concept in physics is valid only if it can be used to explain faithfully the experience we encountered in the physical world. This is the reason why we need to define the direction of angular velocity in the way that we use now.

(Note: you may use "Left Hand Screw Rule" instead of "Right Hand". This also can explain the physical phenomenon. It is only a tradition that physicists prefer Right-Hand more than Left-Hand. The situation is similar to that in linear motion, where we usually define upward motion as +ve. But it is equally good if we define downward motion as +ve. The physical results so obtained do not change.)