Wittgenstein found that games have no one defining attribute that all games share. This fueled his attack on essentialism, the view that there must be something common to all instances of a concept. Because the concept is really a thousand different concepts, the search is fruitless. Once you understand that, you may be spared an argument about nothing. Many practices share resemblances, but there is no one thing that runs through them all.
Language has no essence but only different phenomena related in various ways.
Wittgenstein did not claim that all concepts are family resemblance concepts. As Aristotle put it, one should not treat any subject with greater systematicity than it allows. Of course, it is dogmatic to deny uniformity when it exists. And the processes of naming the stones and of repeating words after someone might also be called language-games. And we extend our concept of number as in spinning a thread we twist fiber on fiber.