"We'll need a safety concept both for the ground"
Think about it... without a working active stabilization system and a carefully engineered for re-entry shape, "stuff" that re-enters usually doesn't make it very far in the atmosphere before being turned into dust.
We have a long way to go until an elevator can be as large as a dino-killer asteroid. Existing designs have spectacular surface are to volume ratios, there won't be much left...
This is, however, a very serious problem on the moon or perhaps mars. And because the moon's gravity is so low compared to the earth, we're almost certain to have elevators all over the moon before the first one on the earth. A moon-vator is so small and light we don't even need new tech to pull it off, just drop a couple billion and we can do it within perhaps 3 years if done privately. Maybe 30 if NASA runs it.
So yeah, its a problem, but not for legacy earthlings.