Therefore, a payload moving up the elevator would either need to be accelerated laterally somehow, or it would bend the elevator over.

If the cable is not anchored to the Earth, this causes the cable to slow down, and wobble a bit since the force is applied sequentially along the cable's length, rather than always on the center of mass, though tidal effects will tend to damp the wobble and keep it oriented practically straight up and down. For any one payload, the effect is negligible, as a space elevator cable would be freakin' massive, but without some sort of active stationkeeping it would eventually come down.

If the cable is anchored to the Earth, however, as soon as it begins to be pulled over even a little bit by the rising payload, the lateral force is transmitted to the ground. The cable pulls on the Earth and the Earth pulls back, re-accelerating the cable with no need for active stationkeeping and ever so slightly slowing the rotation of the Earth. Since the Earth is Gigantic and has ridiculously large quantities of angular momentum and rotational kinetic energy, no one will ever notice this loss in practice.