I posted this on www.bautforum.com with some minor changes. Please rebut if I am in error.
The space elevator won't work if the counter weight is at GEO orbit. It needs to be higher than GEO = perhaps 84,000 kilometers = twice GEO radius. The counter weight travels twice as fast as the GEO point on the tether, so it pulls the tether taught by centripetal force (centrifugal if you prefer)
If the lump = counter weight is low mass and the tether is tapered = more mass near GEO, the total radius can be as much as 336,000 kilometers = 8 times GEO radius = just barely clears the Moon. This means the counterweight is traveling 8 times faster than GEO orbital speed = ideal for getting to Mars and other planets fast with no propellant except to slow down at the destination, and for mid course correction.
Earth's gravity is reduced by about 21 squared = 441 times at an altitude of 84,000 kilometers, so the gravity only opposes the centripetal force slightly. The moon and sun gravity are almost as significant, so they do need to be considered. The elevator is unstable, if it only pulls on the Earth anchor a few hundred kilograms. One design is thinking 20 tons average pull for a space elevator with a 20 ton climber gross weight. Even so it may occasionally be prudent not to launch another climber when the the sun and moon will be at max and minimum counter weight effect while there are climbers somewhere on the tether other than near GEO altitude. The climber is weightless at GEO altitude. The climbers can manage the transients at least a little by accelerating or decelerating at the appropriate instant.
There are more complications. The tether behaves like an extremely long period bungee cord = It starts to stretch as soon as the 20 ton climber is attached, but the stretch transient does not travel at the speed of light; perhaps only a few hundred miles per hour, so a fast ascending climber can out run it's stretch transient producing some effects something like supersonic aircraft. Perhaps this is one reason they are thinking not much faster than 200 kilometers per hour for assent speed of the climbers. Faster would be nice as it takes 420 hours to reach the far end at an average of 200 kilometers per hour, a distance of 84,000 kilometers. Besides the stretch transients, which may persist for weeks, there will be horizontal transients, such that the tether is not a straight line. These will be used to dodge the tether around satellites and space junk which will occasionally be on a collision course. Some have supposed a large space station at GEO altitude: But my guess is all kinds of transients will shake the GEO altitude station if it is attached to the tether, possibly over stressing the tether. As I see it the payload is released from the tether at about GEO altitude and needs to travel several miles in free fall to land safely on the GEO station. Typically the tether is called a ribbon. www.liftport.com has a great forum on the space elevator and related topics. Neil

Perhaps a better first step would be a several mile long tether in GEO.  This could be used to test climber designs among other things.

It may not be better overall and certainly other things, as you suggest, could be tested in LEO with the same equipment.

I do like the idea of GEO testing.  For one thing it would be easier to watch continuously since it is always in the same location relative to the Earth.

IIRC Liftport's roadmap does call for LEO testing before GEO testing.