And once we have one of those, mind as well build a mass driver at the top so rockets don’t have to carry so much of their own mass.

You wouldn't even need a mass driver. You have to build your space elevator so that it's center of mass is where you want it to orbit. Logically, this needs to be at geostationary orbit so that the end point on the ground stays in the same spot. That means you can extend the other end of the elevator to twice geostationary orbit. Lift a mass from the ground to the far end of the elevator and just let it go. It will be flung away out of earth orbit because it'll already be moving faster that orbital velocity at that height. You're limited in the direction you can fling it because it will be flung off by the Earth's rotation, but you don't necessarily need a mass driver.