A space elevator would offer a direct connection between the surface of the Earth and geosynchronous orbits. While the concept has several severe problems, according to my understanding of orbital mechanics, it would constitute a collision hazard to any satellite, not in geosynchronous orbit. Besides the political implications of possibly vaporizing or moving the satellites of major powers, how bad would this problem be? Does it spoil all low satellite orbits, or would it just result in very occasional collision problems? Especially future low-orbit satellite constellations like Starlink seem very much poised on a collision course with a space elevator.
In 2D, it certainly looks like a space elevator will intersect every low Earth orbit.
In 3D, orbits have plenty of space to pass over or under the elevator.
Most satellites are launched into orbits with some inclination relative to the equatorial plane, so they only pass through said plane twice per orbit. If the elevator swipe isn't located exactly at these points at exactly the same time as the satellite, then no collision.
This will happen though. Many objects orbiting Earth have no manoeuvring capabilities (like junk), and left on their own their their orbit slowly changes from tidal effects and atmospheric interaction. Sooner or later the line of apsides of some of them will line up with the current location of the elevator.
So you can still have LEO satellites, they just need to be able to get slightly out of the way (a couple of hundred meters north or south out of the equatorial place should do it) once in a blue Moon.
In a way, this is not very different from how collision avoidance between satellites has to work. Most of the time there's no risk of collision, and when there is a tiny avoidance manoeuvre takes care of the problem. This would not work for space junk though.
Given the monumental effort required to build a space elevator, cleaning up every LEO and MEO satellite would be a small task. Nor will it be a large loss since these satellites can be easily replaced once the elevator is in place.