There's an inconsistency in the preamble. One has to be very careful when converting a quote structure to a report structure:
The book doesn't say, "What's the wisdom behind education?"
Changing this to an indirect question becomes the following:
The book doesn't ask what the wisdom behind education is.
This conversion largely preserves sense, but even now, the reported version may be an inaccurate statement while the quote version may be an accurate if pedantic one - the actual wording in the book might be "What is the wisdom behind education?", for instance.
Addressing the problem about the grammatical (rather than semantic) correctness of various report structures, the tendency is for much less rigid rules to be deemed necessary for both the wording and the punctuation, with the proviso that accuracy be maintained and ambiguity be avoided.
For instance, I've come across (on the web) examples of the use of bid and wish both as reporting verbs proper and quote verbs:
He bade us a fond farewell.
He bade us welcome.
... bade us "Goodbye" [unusual]
She wished us a merry Christmas.
She wished us Merry Christmas.
She wished us "Merry Christmas."
There may be other verbs used in both capacities.
(Say appears to be, of course:
He said {that} she was going shopping.
He said "She was going shopping."
but the meaning changes slightly.)
With the examples given, I'd say that
The book doesn't say what's the wisdom behind education. or The book doesn't ask what's the wisdom behind education. are clumsy elisions, better expressed as The book doesn't say what the wisdom behind education is. or The book doesn't ask what the wisdom behind education is. (the difference is now obvious).
However, "[She] doesn't say what's really on her mind." is quite acceptable grammatically, in fact the accepted idiomatic expression. Though I'm struggling to think of the related quote version.