The problem comes in when you want to write a search function for the game tree. Your search function will make assumptions about how the tree is constructed, i.e. assuming that the children can be produced given only the current board and player, which isn't true for all games.
Your search function will wind up actually having two responsibilities, that is 1) creating the tree and 2) traversing/searching the tree. If you want to do anything else, like add pruning functions (maximum depth, for example), then your search function will also wind up having to manage that.
The problem is, a search algorithm such as breadth-first or A* really has no reason to need know any of those things. If you passed the search function a pre-computed tree already pruned (i.e. capped at a max-depth) the resultant search function would do one thing: search. It would have no need to prune or compute the tree, as that was already taken care of.
This lets you re-use that search function for many other use-cases, rather than tailoring it to a specific game or use-case. In a lazy language, you can pass the search function an infinite tree, and the search function doesn't need to know that it is infinite. It can treat it just as though it's already been computed. Similarly, you can do the same thing with pruning the tree before passing it to the search function.
The end result is that rather than having to intertwine all of these unrelated concepts (generating the data structure, pruning the data structure, and searching the data structure), a lazy language makes it extremely natural (and in fact the default) to write programs where these building blocks are separate entities that can be composed with other pieces of code.
Hopefully that helps make sense of why laziness is a requirement for truly modular code. Note that you can always emulate lazy evaluation with closures (or often, simply using generators), but it's never as natural as just having it be built into the language by default.