The problem is that for a satellite that's above the geosynchronous altitude, as would be the top of the space elevator ribbon, it's natural period is more than a day.
The period of the moon, for example, is 28 days (but you knew that ;-) )
The anchor at the top of this 100,000 km ribbon would have to be orbiting the earth at a rate of once a day, faster than a free satellite at the same altitude, which is what gives it the ability to "lift" the ribbon. If the ribbon were to disconnect, the satellite would immediately go to a higher altitude orbit, in the manner of a rock being swung on a string if you let go of the string.
Even ignoring the tether, going back to the geosynchronous satellite, it travels about 150,000 miles in 24 hours to keep up with the point on the surface of the earth it's above, whereas that point on the earth only travels 25,000 miles in the same 24 hours.
Actually, the space elevator IS in free orbit! The mass of the counterweight above the GSO station is equal to the mass of the ribbon below, so the system as a whole is in balance — a very long, thin structure with its axis pointed at the center of the Earth. Think of the elevator as a satellite 100,000 miles “tall” and you’ve got it.