I don’t think that the Rambam’s letter is from the Geniza (though the Mishnah Berura, unlike the Chazon Ish, does follow kisvei yados). We never assume that a particular Gadol “must” have been aware of the earlier source if he didn’t quote it; otherwise, this klal (i.e., that an Achron who fails to quote an earlier source isn’t followed) would be useless.
The Pele Yoetz, as I have mentioned before, holds that shidduchim requires “a lot” of hishtadlus, even going to another city in order to find a shidduch, and failing to do so can jeopardize one’s shidduch. The Yabia Omer quotes and follows this Pele Yoetz. Even if the Brisker Rav disagreed with the Pele Yoetz, since it is a question of a mitzvah d’oraissa, we must follow the machmirim.
The Brisker Rav is probably following his grandfather’s shitta on bitachon and hishtadlus. However, similar to what I said above, in the Beis Halevi’s lengthy thesis regarding bitachon and hishtadlus, he doesn’t quote the Rambam.