If you want to know about how rav hutner felt about rabbi kook, talk to talmidim. He publicly removed rabbi kooks picture. He changed a ton. Many gedolim did. The truth about him took years to come out.
Rabbi kook wasn’t rav shlomo zalman’s “rebbe muvhak”, i dont know where you got that information from, but it’s not true. I’ve heard that he discouraged people from speaking about him, but did he know the extent of his statements? Who knows. He had better things to do than study controversial rabbinic figures.
Aa for the quotes from rav elyashiv…again, no source. No reason to accept it. The chazon ish said you’re not allowed to learn his hashkofa seforim, they’re assur, they’re treif. That’s in maaseh ish, from many talmidim; rav chaim kanievsky is quoted a lot in that sefer too.
The dati world has tried to make it as if rabbi kook was the gad hador, and admittedly the yeshiva world has erased him from history. Neither are historically accurate. Some gedolim saw through him early on, and he was put in cherem by the rabbonim of yerushalayim over his soccer comments. But many didn’t want to get involved in machlokes. Young rav shlomo zalman just wanted to learn torah, and that he did. Better than almost everyone.