As a talmid of rav belsky, him being my main halacha rebbe and madrich, he never once quoted rabbi kook. He may have said a story or two including rabbi yoshe ber, but never quoted him as a source for anything. What rabbi moshe dovid Leibowitz does is his own business, but it is neither a reflection on the yeshiva or rav belsky. None of the other rebbeim in the yeshiva quote either of the two, at any time, having learned by and having had a shaychus with pretty much all of them.
Additionally, rav belsky was not happy with likut seforim. He did not approve of seforim like piskei teshuvos, and halachikally speaking, even though rabbi Leibowitz quotes rav belsky very often – enough to issue a volume entitled piskei rav belsky, in fact.
Judaica stores are very different from seforim stores. They get wholesale english titles, and you’ll probably see norman lamm and israeli biographies of ben gurion and whatnot too.
I acknowledge that you will find shuls and yeshivos where especially individual bochurim will have the chidushei hagrid series. It’s mostly a shtik, but it does have a lot to do with rabbi soloveitchik’s pedigree, inspite of his degrees.