There are two excellent answers already, but they do not seem to have fully answered the OP's questions.
The book's page
is grammatical but ambiguous unless prior context makes clear which specific page is intended. In the absence of clues from prior context, a native speaker would be very unlikely to say it. It is not idiomatic in most circumstances.
It may mean
a specific but unidentified page in a specific and identified book
any typical page in a specific and identified book
Thus, the phrase is, in the absence of additional context, virtually meaningless and would not be idiomatic.
The book's fourth page is
makes clear which specific page is being discussed. Or
Most of he book's pages are
makes clear that we are talking about a typical page. Or
Some of the book's pages are
makes clear that we are making no claim about all of the pages.
All of the book's pages are
makes clear that every page is intended.
If you are asking whether
The book's page
is confusing, the answer is yes, which exactly why it is not idiomatic except in unusual circumstances. If you are asking whether English has the tools to disambiguate that phrase, the answer is "YES.
You seem to be looking for a difficulty that no native speaker over three years old will ever experience.
EDIT: The fact that this specific phrase is not idiomatic does not make it ungrammatical
The school's playground
is idiomatic on the presumption that the school in question has only one playground. Grammar deals with patterns. There are grammatical patterns in English (as I have pointed out above) to deal with a wide variety of situations involving the Saxon genitive. Being idiomatic means choosing the correct pattern for the situation.