>>>R Kook and R Soloveitchik all learned in Litvish Yeshivas.
This is incorrect
While Rav Kook learned in Volozhin, Rabbi Soloveichick was never formally enrolled as a student in any of the standard Litvish Yeshivas. Not even in Brisk by his grandfather.
Neither of them ever claimed to be following the Mesorah of the Litvish Yeshivos