If the power building the space elevator wants to bomb you, you're going to get bombed.
When the ICBMs go up, early warning radars notice them right away and you still have time to act. Leaders can make it to helicopters and basement bunkers, bomber squadrons can scramble, missile silos can already be empty when hit, road mobile ICBM launchers can still relocate.
But with a large enough number of MRBMs, your opponent might get ideas. They might start thinking about getting away with a decapitation strike.
The military space elevator is more like an ICBM in this case. There will be ample warning when somebody drops something from geostationary orbit (and also when somebody drops something from lower up).
And it's why we have been so worried about Russian nukes--they have used liquid fueled birds, they can't be held ready to launch. Such birds are pretty much only useful for a first strike as they won't be able to launch them once incoming missiles are detected unless they're being held at launch ready (and they can't do that for too long.)
Because that's all they can do
Snipping off just the first few kilometers is not catastrophically destructive yet, and cutting it down further up would require multistage rocket designs, sophisticated steering/targeting and potentially significant yield (you'd need to cut unobtainium, after all...). If you can build a space elevator, you can defend against those.
You better thoroughly inspect what cargo you put on the elevator itself, of course.
