Rav Bick ZTL, Rav Henkin ZTL and others all disagreed with Rav Moshe on many occasion, and they were entitled. But unless you are qualified to agree with one psak over another, and in the base sense of a prevailing halachic behavior in your family or community, you should follow whoever is the bigger posek (or poskim), and that often would mean Rav Moshe. However, that having been said, there is no such Halachic status as “Posek Hador”. Rav Moshe ZTL was great beyond great, but there is no reason to consider his psakim more authoratative than let’s say Rav Aharon Kotler or the Chazon Ish. There would be no reason, let’s say, to follow Rav Moshe’s shiurim for the Pesach seder than those of the Chazon Ish. In fact, Roshei Yeshiva and Poskim, such as Rav Hutner, Rav Eli Meyer Bloch of Telz, the Debreciner Rav, the Chelkas Yaakov and others, sided with the Satmar Rebbe over Rav Moshe regarding the obligatory size of a mechitzah in a shul, and/or the permissibility of artificial insemination, which were the two big disagreements that those Gedoim had in halachah. It was indeed Rav Hutner who approached the Satmar Rav asking him to write a refutation to Rav Moshe’s psak about the Mechitzos. You will not find the phrse “posek hador” anywhere in any meaningful way. The Tzitz Eliezer uses it all over the place in his titles, and, I believe, either the Teshuvos Maharshal writes it among the titles to the Ramah, or the Teshuvos Ramah about the Maharshal. But in any case, the title connotes no halachic status.
Unfortunately, many of those who use Rav Moshe’s psakim do so only when he is maikel. He permitted Cholov Yisroel (only b’shas hadchak – though they don’t pay attention to that part of the psak); he lowered the height of the Mechitzos — psakim such as these made life much easier for the Modern Orthodox, and even the out-of-town Orthodox communities. They believe they need Rav Moshe’s psakim to facilitate their mission as Modern Orthodox rabbis, or to be able to cater to the not-so-frum and do Kiruv. That is not a bad thing. A psak is a psak. However, when the same Rav Moshe prohibits Shabbos clocks (in most cases), or prohibits going to college, or paskens unequivocally that boys are prohibited m’doraisa to be “just friends” with girls, the same rabbonim with “Rav Moshe’s mechotzos” and cholov stam suddenly rely on “other poskim” (though in the case of boys being friends with girls, there are no poskim of anywhere near that stature who disagree with Rav Moshe). Part of it is due, too, to the fact that, at least in America, the other two personalities who were considered Gedolei Hador of that caliber were Rav Aharon Kotler and the Satmar Rav ZTL. Because of Rav Aharon’s stance on college and secularism in general, and the Satmar Rav’s stance on Zionism, there was no way in the world that those two Torah giants were going to be considered authoratative in what constituted the Orthodox community in America in those days. Instead, Rav Aharon was largley ignored, as it was predicted the followers of his hashkofo would become “mere tourist attractions” (thats a quote from Rav Y.B. Soloveichik in his “Five Addresses” about who he refers to as “seperatist Orthodox”. Rav Aharon was the leader of that Hashkafa), and the Satmar Rav was passed off as extreme by these people. In other words, it was “safe” for people to accept Rav Moshe and ONLY Rav Moshe because once you accept someone’s psakim in hilchos shabbos and kashrus, for example, you are forced to at least think about considering the fact that their stance against college or Zionism comes with as least as much authority. Of course, Rav Moshe Feinstein ZT”L deserved all the honor and respect that he received. He was a Gaon among Geonim and a Tzadik among Tzadikim, and one of the great Halachic authorities of our times. Thats not the issue. The issue is the fact that people pick and choose which Gaon-among-Geonim to follow when and because it is comfortable for them to do so.
The Gedolim in the days of the Shulchan Aruch and shortly thereafter have agreed to accept the psakim of the mechaber (and the Rema) as authoritative. The Shach writes that one cannot even claim “kim li” against a psak of the Shulchan Aruch. This is akin to accepting someone as your “Rebbi”, where you follow his psakim. This is the same thing that happened when, let’s say, Klal Yisroel decided that the period of Chazal has ended after the 7th generraiton of Amorayim (Mar Zutra, Mar bar Rav Ashi, etc), and nobody from here on in can add to the Gemora. There was no “halachah lmoshe misinai” that told us that the Gemora was sealed; it was the accepted reality told to us by our Gedolim. The same thign applies to accepting the Shulchan Aruch and Rema.