You can watch the progress of a game without playing in the game.

You can do this by navigating to the game URL of the game that you wish to watch .eg

https://elephantchess.me/game/42b1eb62-f79d-4b36-aaae-33b589bbccd8

The parts here the same as described in Joined Games

Note that the center of the move indicator is a white circle to indicate that you are not a player