I'm not sure 'purpose' is the right word here. Truly what is the purpose of a game? I don't think there's a trivial answer to the question, and I think 'story' is more fair. People care about stories. Stories generate meaning.
I can't speak for anyone else but for me purpose, if I distill the meaning, means that I walk away from the game richer.
Eg
I have experienced an interesting emotion due to a story and the characters. Or I have realized a principle thanks to an interesting strategy (happened to me lately with Defense Grid - I realized I shouldn't build the longest path possible but maximize the times an enemy walks around a defense tower; this translates directly into books I'm reading now about featuritis and user experience - minimize the number of features, maximize their usefulness).
I'm not a native speaker so maybe there's a better word for it that eludes me.
Edit: Story is definitely one (maybe the most important) cornerstone of purpose!
