The statute of limitations applies separately to each instance of infringement, it doesn't start ticking at the first instance of infringement like you suggest.

For example if I hosted a copyrighted movie on my website for 3 years and happened to not get sued for it that doesn't mean I am in the clear to keep hosting that movie forever. Each time I make a copy of the movie to respond to a request is a separate instance of infringement and has its own statute of limitations.

So Microsoft could still be sued for all the copies of the game they have distributed in the past 3 years.