Cherrybim; To clarify;
You posted; “Years ago EVERYONE, including Rav Gornish, accepted and used the “Big Five” hashgachos”………..”especially since his Yeshiva and his family did”
#1 Where did you get this information from, how do you know it’s all true?
#2 What are these “Big Five” hashgachos”
#3 If the Rav once relied on them, those in charge of them were most likely not the same people overseeing those hasghachos now.
Perhaps the Rav does not rely on those who over see them now like he did with others he may have known personally and relied on his personal knowledge of their integrity.
Why can’t this or something similar be true?
You posted; “…so we have every right to assume that Rav Moshe did also, especially since his Yeshiva and his family did”
Where does Shulchan Aurich state clearly that we have the right to make these assumptions that you say, we “have the right”, to make?