Port game example in Beam repo to use InjectorUnboundedSource

[kennknowles]

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. Original Jira may contain additional context.
Reported by: kenn.