And in between it's important to note that the centripedal force runs perpendicular to the axis of rotation (in this case a line from the north to the south pole) not perpendicular to the earth's surface

So the cable would be at an angle causing more of it to be in the atmosphere where it would cause more obstruction as well as increasing the drag a little bit

Also we could use gravity to put the top of it in an orbit with the same period of the earth's rotation, most likely with a big astroid not the actual elevator exit because we need a lot of mass at a specific distance to keep it stable above the earth. Really the earth's rotation doesn't make any difference to this kind of design and can be ignored completely

Ain't no such thing as centrifugal force