I would like to clarify/explain R' Shachter's remarks about the godol hador because I think people have misunderstood what he meant.
RHS was defining Godol Hador in a pure halachic sense. This is part of the gadlus of RHS (following in the footsteps of the Rav), the ability to take things that are seemingly not halachic and to show that it really is a halachic concept and to explain the system of how it works. For more on this see the first piece in Nefesh Harav. RHS said that based on Tosafos Berachos 31b we see that there is such a halachic concept of godol hador. RHS understood that it is based on the well known halacha of rebbe muvhak and that is how it works. Today the term has been borrowed and is not used in it's pure halachic sense.
There hasn't been a halachic godol hador for well over a thousand years, probably not since the time of R' Akiva. The Rambam was certainly one of the greatest poskim ever and yet even in his lifetime many (if not most) of world jewry did not follow many of his psakim. It has been like this throughout the galus. Every community/country has had it's Gedolim/Poskim who they followed. There hasn't been 1 posek who everyone followed. How many people paskened like the Gra when he was alive? A very very small minority. R' Chaim Ozer represented the Lithuanian/Poland yeshiva world. He didn't represent the Hasidim, they followed their Rebbeim, he didn't represent the Hungarians they had their poskim. As RHS said this is the way it has always been.
Based on the above, it is clear that R' Elyashiv is not the godol hador in the halachic sense, he is the posek for the Lithuanian/Ashkenazi community in Israel and is a tremendous Gadol B'Torah. That however, does not make him "the Godol Hador" in a halachic sense.
I think it is safe to say that there won't be a godol hador in the halachic sense until Moshiach comes, there are too many different approaches to halacha now. You have the chassidish approach, the sefardim, and even among the misnagdim you have those that follow the Mishna Berura's approach (see Dr. Chaim Soloveitchik's article Rupture and reconstruction) and those who follow the Gra/Brisker approach. One posek cannot encompass all of this.
What comes out of all this is that a person needs to have a Rav/Rebbe who is qualified to have an opinion and follow that persons psakim. The fact that there are other poskim who disagree shouldn't bother you. Every posek needs to take things into account, if R' Elyashiv disagrees with their opinion they need to think maybe they are wrong, but if they think things through and believe that they are right then they are obligated to state their opinion and follow it.