Far as I can figure it, the kinetic energy is small compared with the potential energy.
For a one-kilogramme object on the equator at the base of the tether, it is already doing 40,000 km per day. At a geostationary height of 36,000 km, it would be doing 226,000 km per day. So it changes from 416 metres/second to 2616 m/s, which I make 2.42 MJ.
The potential energy of gravity, integrated from surface to 36,000 km, seems to come out at 37.7 MJ, so the potential energy outweighs the kinetic by about 15:1. (I have to admit my integration is a little flaky.)
I agree the tether will slightly lag the planet, but then the tether has a large weight outside the geosynchronous point to provide stability through centripetal tension. I would take the view that the kinetic energy is taken from the forward pull of the planet when objects ascend, and this is recovered when they descend. It would even appear that the tension due to the lagging tether would tend to pull the elevator up the tether.