Yeshivish rockstar:
It’s interesting that rather than praising Rabbi Shapiro’s sefer, which is trying to uproot the A”Z of Zionism from Jews the world over (who have been influenced by that idol, R”L L”A), you prefer a book that contains clear references to, and themes of, A”Z. Rabbeinu Yonah’s vort on “ViIsh kifi mahallalo” comes to mind; you know what someone is about based on whom/what they praise.
Regarding the Kook-shmook story that Rabbi Shapiro quoted, he said he heard it from the original source. This story was also said over in Brisk by Rav Berel. It is well-known in those circles. So Rabbi Shapiro very obviously did not make up the story.
But you don’t need to believe or not believe stories. You can instead open up some sefarim. Both the Satmar Rav and Rav Elchonon HY”D blasted Rabbi Kook in incredibly strong terms. Each of the Satmar Rav and Rav Elchonon applied a different Rabbeinu Yonah, including the one I quoted above, with Rav Elchonon concluding that Rabbi Kook was, therefore, a “rasha gamur”, as Rabbi Shapiro quotes.