Rav Moshe was no fan of the Medinah. He was a talmid of the Brisker Rov and - while a fiercely independent thinker - subscribed fully to the very, very negative attitude that his Rebbi had to the secular establishment.
I remember a shiur he gave 20 plus years ago where he expanded on the gemara [Megilla 6] that says that a rasha swallows up a tzadik who is greater than him but not a tzadik gamur. When it comes to a tzadik gamur, a rasha has no power. He didn't say it explicitly [he rarely spoke explicitly about current events but was ALWAYS talking about current events. His focus was trying to show how the Hand of Hashem is guiding all of the events and what our job is] but alluded to the fact that anybody who is part of the secular framework and is connected to the secular establishment, even if his intentions are sincere, will be swallowed up by them. But if one is COMPLETELY DIVORCED from politics, government, etc. etc. and clings to Hashem - then the reshaim will have no power over him. This is in stark contrast to the prevailing attitude in many circles that we must be full partners with other Jews in building the State of Israel. He was no fan of Statehood and all of its trappings.
Was he right or wrong? In order for one to argue with him - he would have to be on his level [i.e. Know kol ha-torah koolah nigle and nistar - bi-iyun, have his level of Yiras Shomayim etc. etc.]. Since I am not - I will not open my mouth [or press my keyboard].
So on that level he was light years away from the Mercaz Harav world.
HOWEVER, there was an area where he was much closer to that world and very far from the world he came from. He spent his life talking about EMUNAH. Understanding HASHEM. Talking deeper, deeper and yet deeper about אלוקות. In the Yeshivos they learn and learn well, Baruch Hashem keyn yirbu. But there is something missing. It seems that the same level of depth that is found in the study of Bava Kamma and Zevachim is lacking in the study of אנכי השם אלקיך and את ה' אלקיך תירא etc. etc. There is very little emphasis in the areas of Emunos Vi-deyos.
In the last hundred years in the Litvishe world there was Rav Dessler, Rav Hutner and a few others who were great teachers of pnimiyus ha-Torah, but all in all, the stress was always on gemara and Emunos Vi-deyos fell by the wayside. Rav Moshe came and taught all of the chasidishe sefarim like the Bnei Yisaschar, Sfas Emes and Rebbe Tzadok without ever mentioning their names. In addition, he taught [by name] Toras HaGra, Ba'al HaLeshem and others of the non-chasidic persuasion זיע"א. He received a LOT of flak for his shiurim including by no less a personage than HaRav Shach ztz"l who didn't think that סתרי תורה should be publicly taught [I believe those were the grounds for his opposition but maybe there were other issues as well]. But he wasn't deterred. He felt that it must be done - so he did it. Rav Dessler ALSO was berated by one of the great tzadikim of his generation for teaching סתרי תורה. He politely listened and then politely continued teaching. [Rav Moshe was very close to Rav Dessler and was a talmid of Rav Hutner as well].
That was exactly the attitude of Rav Kook. Emunah and pnimuyus hatorah must be taught. It is not enough to learn שור שנגח את הפרה but one must learn וידעת היום והשובת אל לבבך כי השם הוא האלקים בשמים ממעל ועל הארץ מתחת אין עוד. To say that Rav Kook was criticized would be an understatement. But he felt that his task in life was to spread דעת השם and wouldn't let anyone stop him. See the sefer אורות הקודש where this is discussed at length.
So there is a very interesting convergance between the Charedi-Brisker-Rosh-Yeshiva and the Dati Leumi world. לימוד אמונה בועמק! Rav Moshe's roommate in Chevron was Rav Yeshayahu Hadari Shlita who offered him the job of Rosh Yeshivat Hakotel but Rav Moshe wasn't interested. It is a good thing, for had he taken the job - Hakotel would have been much better off but then the Charedi world would have shunned him and lost out on all of his teachings.
This emphasis on limmud pnimiyus hatorah is a huge focus in Chabad as well. Rav Moshe was NOT A FAN, at least of the modern Lubavitch movement [at least not the Meshichist aspect]. But again - on this point he was completely on their side and against the prevailing Litvish attitudes.
So let us continue on his path and constantly delve deeper and deeper into sugyos of Daas Hashem and just as we can learn a Tosfos for weeks [if we go to certain Yeshivos] certainly, a fortiori [קל וחומר] we should spend that much and much much more effort in understanding Hashem.