Well it's anchored on the moon at the other side

It's dangling down from the moon. Going at moons speed - 28 days per revolution instead of geostationary 1 day. So less centrifugal force form rotation that is compensated by the cable strain.

The end is at that height. Its center of mass is much higher, and it's rigid enough to hold together. It's also not orbiting—its other end is nailed to the lunar surface.

Well I started thinking about it more.

If it's orbiting at the speed of the moon, it's going slower than geostationary orbit speed. So its tendency will be to fall towards the earth. But since it's attached to the moon, this won't happen.

Not sure how much station keeping would be needed to hold this in place.

Also will have to deal with collision risks with objects in geostationary orbit. Though I don't think geostationary orbit is a particularly important factor. Could probably be a few thousand Km above geostationary orbit without any significant adjustment to their plan.

It is not a point. Consider two masses connected by an rope, one in an higher orbit. The entire contraption will move along their common center of mass, and there will be tension on the rope. The same thing applies for an lunar elevator, one end is somewhere close to geostationary orbit, the other is the moon, and the entire system moves as a combination of internal tension and the movement of the center of mass.

The tether is hanging from the moon, not orbiting earth. There is tension in the tether. It's not in free fall.

because it's attached to the moon, which is preventing it from falling into the earth which it would otherwise do at that altitude going that slowly.

And the Moon orbit is not a perfect circle, the Earth-Moon distance varies quite a bit (about 40000km between perigee and apogee), so to actually keep the cable at the same earth altitude you will need to change its length quite a bit (seems like around 125km/hour).