None of this should be construed as an endorsement of space elevators, though. They will never happen on Earth as it exists now. Over and above the material science, you have to clean out LEO of all satellites before you even start construction. It’s just a really dumb idea all around.
https://arxiv.org/abs/1505.07424
You build an orbital ring above the altitude of the LEO satellites, use non-synchronous skyhooks to pick up payloads from inside the atmosphere at an altitude that can be efficiently served by aeroplanes, and then fling them the rest of the way to geosynchronous or interplanetary transfer orbits by accelerating them along the ring.
A satellite impacting the tether will definitely "stop" it, in that it will, at the very least, melt an impact crater into it if the cable is wide enough to not be cut. It is unlikely that the cable will be that wide. I feel like a broken record, lately, but the dominant concerns of impact modeling at orbital velocities are (a) mass and (b) energy. Everything traveling at 8 km/s effectively splashes into whatever solid object it hits, both because the room temperature shear resistance of the materials involved is orders of magnitude less than the shears involved, and also because the kinetic energy is dumped into thermal energy, melting or vaporizing the materials involved. One tends to get results like: (https://www.esa.int/spaceinimages/Images/2009/02/Hyperveloci...) That link notes that pressures of 365 GPa are reached, and typical yield stresses of theoretical nanotube cables are around 100 GPa.
A carbon nanotube cable is unlikely to be more than an inch or two across at LEO. You need that kind of strength to beat the space-elevator-equivalent of the rocket equation, which governs the taper needed to get the cable to even support its own mass in Earth's gravitational field.