“In this context, it means (at least according to most people) that we all have to listen to what they say when they make a pronouncement for the klal.”
I don’t know of any source for this idea. There is a concept of Rabban Shel Yisroel: a teacher whose piskei halacha are excepted by all of Klal Yisroel, but I cannot think of anyone who would fit that standard since the Chofetz Chaim (maybe). One should follow the psak of a Rabban Shel Yisroel in circumstances where one can (a) not determine the halacha for oneself; and (b) does not have personal Rav who has a ruling on the issue or case.
Outside of the above, while it might be wise to listen to the pronouncements of great Talmidei Chachomim and Tzadikim and one should prudently re-assess their personal positions in light of such pronouncements, there is no issur in disagreeing or not following such pronouncements.
Whether or not someone is a great Talmud Chachom can be determined by reading their works, listening to their shiurim, and seeing how their opinions are treated and discussed by Talmidei Chachomim you know personally. But once you are dealing with a signficant level of Talmud Chachom it is very difficult to distinguish between them. Is the Rambam “greater” than the Ramban or the Ran?
Tzidkus is even more difficult to determine. Gadlus (a term which I view as a combination of Tzidkus, Chochma, and influence on the klal) is something that is extremely difficult to determine without first-hand experience. The best you can usually say is person X is “considered” or “treated as” a Gadol. It is certainly true that Rav Moshe Feinstein was treated as a Gadol in a way that Rabbi Avigdor Miller never was. It is also true that from their works, Rav Moshe Feinstein would appear to me to be a greater Talmud Chachom. But I never gave them a bechina, I don’t know what was in their hearts, and I can’t judge their ultimate positive influence on Klal Yisroel. Furthermore, as Kapusta wrote, there is no constructive purpose to such a discussion.
Now the way I normally understand the idea of Daas Torah (as opposed to psak halacha), is that when making life decisions the outlook of a Torah infused mind will have insights as to the ramifications of those decisions that a less-Torah infused mind might miss. And so, before making serious life-decisions, it is usually worth your while to consult with Torah sage that knows you, that understands your personal and societal situation, and can give you advice from a Torah perspective. Rabbi Avigdor Miller is someone whose life was devoted to learning and teaching Torah. He was Shul Rav and was therefore regularly involved with the lives and issues of ordinary frum people in America. In my opinion, he is exactly the sort of person one should consult to get Daas Torah on an issue.