The Gra did not recite ברוך הוא וברוך שמו because it may prevent answering amen (Maaseh Rav, 43) or because he considered it to be a hefsek (Tosefes Maaseh Rav).
The Shu”T Rosh (4, s. 19) states that his father said ברוך הוא וברוך שמ at each brocha he heard:
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The Torah Temima (Devorim ch 32, s”k 36) cites the Chareidim that relies on a medrash that requires saying ברוך הוא וברוך שמו at each brocha, leaving him wondering what Ashkenazim rely on to omit saying it, then cites the GRA. (see 4 lines from bottom of page):
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