In Nefesh Harav, Rabbi Shachter writes that the Gr"a (and subsequently Rav Soloveitchik) did not say מזמור שיר חנוכת הבית לדוד before davening, as ברוך שאמר is a מתיר to say Tehillim (specifically when it comes to the words "ובשירי דוד עבדיך," etc.):
This got me thinking about the use of Brachos. If we follow the opinion of R' Yosef Dov Soloveitchik discussed here, if we use brachos as a "matir," how can we say Tehillim during the day (at least not without the words "ובשירי דוד עבדיך" - or something of the like)?
Perhaps this idea is more similar to Birkas HaTorah than to other brachos (like birkas hanehenin), as it lasts all day. Maybe that can answer why one may say Tehillim throughout the day as well as the Ashrei before Mincha.
Which leads to another question: How did the Gr"a and the Rav recite Selichos (a service that begins with Ashrei, a chapter of Tehillim) in the morning - before the recitation of ברוך שאמר? Do the words "תהלה לדוד" act as a matir? Would that answer our Selichos question, allowing for chapters of Tehillim like Ashrei to be said before or after ברוך שאמר? (This leads to yet another question: is ברוך שאמר only a matir for non-"תהלה לדוד" Tehillim, like the Hallelukas?)
Is it possible that I am overthinking things and that the Nusach of the Gr"a and the Rav is merely a midas chassidus that is by no means required?