This icon in Apple Watch Control Center

Yes.

It's the only place where the AirPlay Mirroring options can be accessed. Of course, AirPlay can also be accessed from the the player controls while playing a movie.

Usually when that page is stuck, you simply need to power off the watch.

Press and hold the large button (not the crown) and slide to power off. It's slow, so give it a minute or two to shut everything down and then power up the watch.