I have two general points to make; A) possible couterexamples, and B) Wittgenstein's point.
A) I think that it is helpful to think of the difference between work and leisure and how this relates to games.  Under your definition there is nothing I can see that rules out work from being a game - 'value' can be understood so broadly.  Also there are times when playing football can be a leisure activity and times when it can be work - consider professionals.  The 'value' involved would be markedly different but we would still call them both games.  So how would you make the distinction?  Working in a bank is definitely not playing a game and yet the professional footballer may be just as if not more serious about his job and may receive little or no pleasure from it other than the paycheck.
B) When we consider the context of the Philsoophical Investigations, we realise that Wittgenstein was not as concerned with a full proof account of family resemblance as he was with showing an understanding of language based upon usage - the "grey areas" are a consequence of the vagueness and changing usages of language in different contexts and over time.  What Wittgenstein was trying to show was the importance of the activity (game) of language and the practices and world that are inseparable from it.  Further, that philsophy is not about doctrines and theories and definitions (thinking abstractly) but rather about seeing the activity of language (looking at particular situations of language games).