The top of an ORS isn't in orbit — it's stationary with respect to the ground. You still have to burn the Δv to get into orbit. The idea is that it's easier to go slowly up, where the natural speed you're at will (eventually) match the Earth's rotation.

I'm not sure which structure you're thinking of, but the first sentence of my link was this:

"""An orbital ring is a concept of an artificial ring placed around a body and set rotating at such a rate that the apparent centrifugal force is large enough to counteract the force of gravity."""

The tether isn't in orbit, but the ring absolutely is. And you can exchange momentum with the ring to reach orbital speed.