It doesn’t need any sideways velocity, because the elevator’s center of mass is located at the geosynchronous altitude, where the orbital speed relative to the surface of the Earth at the Equator is zero.
I don't think that's right.
An object on the surface of the earth has a "sideways" velocity relative to a fixed frame of reference of slightly in excess of 1000 mph.
When it gets to the top of the 100,000 km ribbon, on its way to Mars for example, it's got a speed relative to that same fixed frame of reference of roughly 17,000 mph.
Where'd that other 16,000 mph come from?
Using a conventionally-launched orbiter, the launch rocket climbs vertically for a short period of time, but quickly rotates off the vertical to start developing the 18,000 mph necessary for low-earth orbit.