Port game example in Beam repo to use InjectorUnboundedSource

We developed an UnboundedSource so users didn't have to run a separate injector, but it isn't in the Beam example. We should put it there, which eliminates a big chunk of complexity.

Imported from Jira [BEAM-1785](https://issues.apache.org/jira/browse/BEAM-1785). Original Jira may contain additional context.
Reported by: kenn.

Start by locating the Beam game example and its current injector setup, then inspect InjectorUnboundedSource and the additional context in Jira BEAM-1785. Done means the example uses InjectorUnboundedSource directly and no longer requires a separate injector, while preserving the example's behavior.

Written by the indexing model from the issue text.