Unless somebody has changed the rules since I last looked, center handles all enroute IFR (instrument flight) traffic in a larger area, not just that above 10,000 feet. There are significant numbers of remoted radio transmitters and receivers so center can talk to all the aircraft over a wide area. Relocating the whole thing to another area doesn't just mean re-routing the radar replies, and coming up with another fully manned center, it means rerouting the audio and control to all those transmitters and receivers. It's not a trivial job at all.
To give an example of just how far away center can be, I live in far west Texas, about 70 miles from the Mexican border. When I call up center, I'm talking to people in a room in Albuquerque, NM. I believe their remote transmitter for this area may be in Ft. Stockton, TX, about 70 miles from here.
A nasty problem it is, and quite possibly one with no affordable solution.