Softwords,
1. That is not what Rav Moshe said. He was asked by a secular reporter how he became accepted as a gadol and he replied that people asked him questions and liked his answers. They asked more questions and he became known. In fact, while sometimes gedolim tell avereichim to go into the rabbanut (e.g. the Chofetz Chaim with Rav Kook) once they start paskening a consensus develops in the Torah world. Sometimes it happens “by accident”. After Rav Shlomo Ganzfried got married he opened a liquor business (this was a common Jewish business then and there). While he was waiting for customers he learned and wrote. Customers never came and he was more or less compelled to agree to be a the rav of a city.
2. Rav Kook’s positions on Zionism were more from hashkafic and kabbalistic perspectives. Perhaps that is why non-Zionists are quite willing to cite his halachic opinions.
3. While often a pesak is purely halachic (e.g. opening bottles on Shabbat) it is often the result of hashkafa. For example, those who are meikal about Ashkenazim eating in Sephardi homes during Pesach feel that the unity of a mixed community is more important. Those who are machmir feel that communities should be separate (even to the extent of ethnic discrimination in school admissions).
4. Some who call it apikorsut are using the traditional Jewish tool of exaggerated language (lashon guzma). In fact, once Rav Aahron Kotler referred to a certain rav as an am ha’aretz. The Satmar rebbe was very upset until someone said “The rav says ‘am haaretz’ like the rebbe says ‘apikoros'”.