Not to mention that if one state had the right to shoot down a satellite, anyone in the path of a satellite's orbit would then have the theoretical right to shoot down said satellite.Yes and yes.
But shooting down another country’s satellite would be seen as a hostile act, at the very least the US would retaliate by shooting down Chinese satellites. “Freedom of Space” as it’s known has been tacitly recognised since the first spysats went up in the early 1960’s. While they mean that another country can see what you’re up to, you can also keep a close watch on them and make a Pearl Harbour style sneak attack impossible. So the major powers are content to put up with the nuisance because overall it helps to reduce tensions.