A space elevator not only raises payloads, but also accelerates them to geostationary orbital velocities (~3000 m/s). Luckily these velocities are much much low

A space elevator not only raises payloads, but also accelerates them to geostationary orbital velocities (~3000 m/s). Luckily these velocities are much much lower than LEO orbits (7600 m/s). Ideally the energy to perform this acceleration comes from the rotation of the planet below, transmitted via the tether. The anchor station would have to be stabilized to prevent oscillation and rotation. However that technical challenge is minor compared to the tether material itself. Another advantage of a space elevator is that we can use electrical power from a base station instead of onboard power on the vehicle alone. Also the power can be much lower than a comparable rocket. To ascend on the rail/tether you can accelerate much slower. It doesn't need to lift fast or fall. It can just slowly accumulate altitude and velocity.