Correlation is not causation. The popularity and success of games and entertainment are causally related to disposable income availability. They aren’t related to the state of an empire or its decline. The games are also present during the growth of an empire too.

The only relationship between these things and decline is: during decline, the importance of tradition and patriotism increases. If the games are solidly entrenched as a tradition at that point, it becomes hard to cancel them, because doing so acknowledges decline.