Regardless of what one thinks about their Hashkofos, Rav Kook and Rav Yoshe Ber were both Gedolei Olam.
Rav Kook was an outstanding Talmid Chochom who was an expert in all areas of Torah. He authored many works in Halacha including, “Shabbat Haaretz” a work on the Halachos of Shemittah. He also published Shailos Uteshuvos Seforim which adressed questions about almost all area of Halacha. it is quoted that The Chazon ish would recomend Talmidim to learn it. He was also an expert in Machshava and Kabballah as evident from many of his other Seforim.
it is true that he was a zionist, which many view as negative, but is important to note that there are a variety of views on the Zionism and the state of Israel from various Gedolim, and many Great Gedolim viewed The State of Israel as a positive thing. Granted, Rav Kook was a much stronger and more radical Zionist than the other Gedolim, but nevertheless many of the Gedolim did view The State of Israel positively. Rav Isser Zalman Meltzer, Rav Tzvi Pesach Frank, and Rav Ovadya Yosef (in a Teshuvah about saying Hallel on Yom Haatzmaut) all used the term “Aschalta Digualah” about the state of Israel. Rav Dessler was not willing to go that far, but he nevertheless writes that the state of israel is a gift from Hashem (Michtav Meiliyahy). Rav Yaakov Kamenetsky is quoted similarly (Emes Liyaakov, in a footnote on Parshas Bo). Rav Herzog and Rav Zevin, two Giants of the last century, recited Hallel on Yom Haatzmaut. The Ponivitzer Rav, refrained from saying Tachanun on Yom Haatzmaut. (There were certainly many Gedolim who felt otherwise, though).
The Gedolim, with a few exceptions, viewed him as a Gadol in a full sense. Here are just a few examples:
Rav Isser Zaman, Rav Tzvi Pesach Frank and Rav Aryeh Levine, were very close to him and talked very highly about him. Rav Elayashiv at a young age, developed a Kesher with Rav Kook, and Rav Kook was in fact his Shadchan and Mesader Kiddushin. He seems to have been Machshiv him later on life as well, and he even supported the firing of and editor who took out references to Rav Kook’s Seforim. Rav Shlomo Zalman Aurbach was Machshiv Rav Kook very much, and even got a Haskamah from him on his Sefer Meorei Haish. Rav Yaakov Kamenetsky is quoted as calling Rav Kook “A Gaon and a Tzadik” (The making of a Gadol p. 1087.) Rav Chaim Ozer Grodzinsky, in a letter to Rav Kook, praised Rav Kook very highly. Rav Ovadya Yosef quotes Rav Kook several times in his Teshuvos. Even Rav Avigdor Miller who very strongly disagreed with Rav Kook’s ideology, when asked for his opinion on Rav Kook, makes sure not to directly criticize Rav Kook. At one point in the answer, he even says “I want to be very careful with the kavod of Rav Kook.” Most of The Gedolim didn’t agree with much of his Hashkafah, but that in no way undermined the respect they had for him.
Rav Yoshe Ber, as well was undoubtfully a Gadol. Most of the other Gedolim had objections to some of his Hashkafos, but that should not undermine our respect for him.
The fact that he had an openness to secular studies should not be viewed as being due to his falling prey to the Yetzer Harah. The question of how much openness to secular studies is an issue which has been debated for many centuries, with Chashuv people on both sides. (There are obviously some things which shouldn’t be studied. The question is just where to draw the line). Already in the times of the Rishonim, this was a major difference between The Ashkenazi and Sefardi Rishonim. While the Ashkenazi Rishonim generally viewed secular studies, especially philosophy, with disdain, some of the Sfardi Rishonim such as The Rambam and Meiri were very knowledgeable in many secular subjects. in later centuries as well, there were different perpectives. Great Leaders, such as The Chasam Sofer and the Beis Halevi were more limiting in their views of secular studies, while The Rama and The Gra seem to have been more open to forms of it. Later on, it became one of the major differences between the Lithiunian Rabonim and Rav Shimshon Rifoel Hirsch, Rav Hildenscheimer and the other Rabonim in Germany. The Fact that Rav Yoshe Ber was more open to secular studies, therefore, does not have to be viewed as an inherently negative thing. He was following the lead of great people before him (although it was admittedly different that the Brisker Mesorah). It is important to note, also, that when he went to university, it was a much safer place than it is now, and other great people went such as the Lubavicher Rebbe and The Sereidi Aish.
In regards to his wife not covering her hair, it is important to note that in that time period there was unfortunately a big laxity in women covering hair, and many lithiuanian women, including wives of other prominent Rabonim, did not cover their hair. Rav Shach is even quoted as allowing a someone to marry a Lithiuanian girl who didn’t cover her hair, because it was viewed by the women as not a big deal, and did not reflect an overall laxity in Halacha on their part (although certainly it is Assur).
It is true that Talmidim quote conflicting things from him, but this does not make him any less of a Gadol. Much of it has to do with the fact that his Hashkafah changed greatly in certain areas over time. for example, in his early years he gave a Hesped for Rav Chaim Ozer which discussed how Daas Torah is necessary even in totally non-Halachic scenarios, a position which seems to be at odds with his later Hashkafah. Also, he switched from Agudah to Mizrachi. A switch of opinion is totally acceptable and can be found in other Gedolim also. Rav Hutner, for example, went from being a Talmid of Rav Kook, to adopting an almost Satmar-like position on The State of Israel (although he was still Machshiv Rav Kook). Some of it may also have to do with the fact that different Talmidim interpeted his stamements in different views. This is not a Chisaron in Rav Yoshe Ber, it is found with many other Gedolim also. Rav Aharon Kotler, Rav Yaakov Kamenetsky and Rav Hutner for example, were all very influenced by The Alter of Slavodka, but they all had unique perspectives. The Gemarah also is full of conflicting reports from Amoraim (as someone pointed out before). Antignos Ish Socho had two Tamidim who left Yiddishkeit because they misunderstood what he meant in Pirkei Avos (See Avos 1,3, with Meforshim).
Although many of the other Gedolim disagreed with some of his Hashkafah, almost none of them disrespected him and many of them were clearly Machshiv him. Rav Moshe Feinstien called him every Yom Tov. Rav Hutner referred to him as a Gadol. Rav Shmuel Rosofsky was in Boston in the end of his life and would often talk in learning with Rav Soloveichik. The ponivicher Rav is quoted as calling him The greatest Rosh Yeshivah in ASmerica. These are but a few examples.
They were both huge Gedolim and should be viewed as such.