There's a specific hand-turned reamer that cuts only the leade / throat area of a .223 chamber to 5.56 NATO specs. It doesn't mess with headspace because the cartridge neck area is not changed. You just oil it, drop it in and turn it until you feel it turn smoothly (cutting completed).
You can do that. Just don’t try to load 5.56 in a .223 chamber. Pressure Likely it won’t do much. But I won’t try it. Certainly not a wholesale switch. Reloading and k owing what you’re doing is a different story.