Why does angular velocity not change the direction of linear velocity in circular motion?

[Boomzxc]

So we know that an object undergoing horizontal circular motion, has a angular velocity with direction perpendicular to the plane of motion, and also a linear velocity with a direction perpendicular to the acceleration towards the centre of motion

Question : if we sum/add the linear+angular velocity, wouldn't there be a Resultant velocity, at an angle above the horizontal?something like North-West,
Why does the object still undergoes motion along the horizontal circle(west)??

 

[haruspex]

So we know that an object undergoing horizontal circular motion, has a angular velocity with direction perpendicular to the plane of motion, and also a linear velocity with a direction perpendicular to the acceleration towards the centre of motion

Question : if we sum/add the linear+angular velocity, wouldn't there be a Resultant velocity, at an angle above the horizontal?something like North-West,
Why does the object still undergoes motion along the horizontal circle(west)??

You cannot add a linear velocity and an angular velocity. They are dimensionally different. It would be like adding an angle to a distance.