The idle countdown that hides the cursor and the play/pause overlay was only
started from mousemove. A cursor that stayed still over the player while a
click, a key or a scrub release started playback never got a countdown, so
the overlay stayed on the video until the mouse moved again.
Arm the timer from one place and rerun it whenever playback or fullscreen
changes, keeping the cursor-over-player state in a ref so the same rule
applies from every entry point.