It could become a thing.
If it’s any comfort, you’d asphyxiate before the fall killed you.
Assuming one didn't asphyxiate or freeze to death on the way up, unfortunately, if during the day, you'd likely suffer immolation (due to high temperatures in the thermosphere) before losing consciousness from asphyxiation. Also, even if not either, the fall doesn't kill you; it's the impact.
If you crawl only to the LEO altitude (e.g. 300 km) on the space elevator, you will be in space, but not in orbit. If you let go of the elevator there, you will immediately start falling to a gruesome death.