Make RemoteStateMonitor and children non-movable

This turns the runtime error of moving while a monitor has started into a build time error instead.

As a side effect, this fixes tsan errors on the move-ping-state-monitor tests that are no longer neeeded.

Tested 100 runs: http://sponge2/a4c5b2c0-9bb0-416f-80b3-0920813c0246

PiperOrigin-RevId: 866018932
Change-Id: I526b0884a3fe5e6ff8925f8d78e45eb9e3f1681e