Why is the "general notion of a reduction [...] inherent to the notion of self-reducibility"?
I think you may be misunderstanding the sentence "Note that the general notion of a reduction (i.e., Cook-reduction) seems inherent." This is not about reductions being inherent to self-reducibility (in the sense Goldreich uses it), but rather about Cook reductions being inherent to this notion, in the sense that they cannot be replaced by  Karp reductions. Goldreich makes a distinction between a "general notion of reduction", meaning a Cook reduction, and the restricted notion of a Karp reduction. Then he points out that the fact that Cook reductions in his definition of self-reducibility cannot be replaced by Karp reductions is not merely because of some formality (i.e. "syntactic considerations") but is inherent because a Karp reduction only makes a single query to the decision problem oracle, and, hence, only reveals "a single bit of information". The exercise shows that many more than a single bit of information need to be revealed (log many bits, as you point out).