It was an appropriate response since Baruch Shem is also a type of Kabalas Ol Malchus Shamayim. Read below from dafyomi.co.il (or the Pachad Yitzchak in the original – but that is some heavy learning!):
2) SAYING “BARUCH SHEM KEVOD MALCHUSO” QUIETLY
QUESTION: The Gemara teaches that after one recites the verse, “Shema Yisrael,” he should recite the verse, “Baruch Shem Kevod Malchuso” quietly. Rebbi Yitzchak relates an allegory to clarify why this verse is said quietly. He says that it is comparable to a princess who smelled the aroma of delicious food. On one hand, she is embarrassed to ask for the food explicitly. On the other hand, if she does not partake of the food, she will suffer as a result of her urge. In order to resolve her dilemma, her servants bring the food to her quietly without announcement.
This analogy implies that there is some element of embarrassment in the recitation of the verse “Baruch Shem,” and that is why it is recited quietly. Indeed, the NEFESH HA’CHAYIM (3:6) and other Kabalistic sources explain that it is a lower level of declaration of the unity (Yichud) of the name of Hash-m.
However, other sources indicate that “Baruch Shem” is a higher form of Yichud ha’Shem, and not a lower form as the Gemara here implies.
First, the TUR (OC 61) cites the Midrash that says that Moshe Rabeinu heard the angels declare, “Baruch Shem Kevod…,” and he wanted to incorporate it into the prayers of the Jewish people. However, because it is an “otherworldly” praise and is too lofty to be recited in this world, he could institute only that it be said quietly. By saying “Baruch Shem” quietly, we show that we have no intention to encroach on the domain of the angels.
Second, we find that in the Beis ha’Mikdash, the Jewish people would respond with “Baruch Shem Kevod…” in place of “Amen” to every blessing they heard from the Kohen Gadol (Berachos 63a, Ta’anis 16b). Due to its lofty status, “Baruch Shem” may be said aloud only in the holiest place.
Third, the MAHARAL (Nesiv ha’Avodah 7) writes that the reason why we recite “Baruch Shem” aloud on Yom Kippur is because on that day we are elevated to a higher realm of existence. Similarly, Yakov Avinu recited this verse aloud because he was on a higher realm of existence.
These sources seem to contradict the implication of the Gemara here that “Baruch Shem” is embarrassing in some way.
ANSWER: RAV YITZCHAK HUTNER zt’l (Pachad Yitzchak, Yom Kippur 5:2:15) explains that both implications are true and do not conflict with each other. They reflect different aspects of “Baruch Shem.” In one sense, “Baruch Shem” is a lower and embarrassing form of praise, while in other sense, it is a lofty and holy form of praise.
“Baruch Shem Kevod Malchuso…” means that the name of Hash-m is eternal and will remain forever. The name of Hash-m, however, is comprised of two different elements (see Pesachim 50a and Insights there). There is the name as it is written, which emphasizes the eternalness of Hash-m, and there is the name as it is pronounced (the name of “Adnus”), which emphasizes the sovereignty of Hash-m in this world and expresses that Hash-m is the Master of the world. The name of “Adnus” is used only in this world; it has no place in the World to Come, where Hash-m’s name will be pronounced the same way that it is written, as the Gemara earlier teaches (50a).
Accordingly, it is inappropriate to say “Baruch Shem Kevod… l’Olam va’Ed” in reference to the name of “Adnus.” That name is used only in this world, while “Baruch Shem… l’Olam va’Ed” is a praise for the eternal use of the name of Hash-m. The name of “Adnus” is a lower level of Yichud ha’Shem, a Yichud only for this world that expresses the limited extent to which we are able to perceive Hash-m. It does not express the way that Hash-m will be perceived in the next world.
In contrast, when “Baruch Shem” is used in reference to the name of “Yud-Heh,” it is an appropriate and lofty praise, because it means that the name of “Yud-Heh” will be blessed in this world and in the next.
Praising the name of “Adnus” with the verse “Baruch Shem” is a lesser form of Yichud, because it applies only to this world. Praising the name of “Yud-Heh,” the name of Hash-m as it is written, with the verse “Baruch Shem” is a much higher form of Yichud. The angels experience Hash-m’s presence and perceive His reality in the ultimate way, the way expressed by the written form of His name. When they say “Baruch Shem,” they praise that name, the name of “Yud-Heh.” Similarly, in the Beis ha’Mikdash, the people declare “Baruch Shem” in response to the Kohen Gadol’s elocution of Hash-m’s name as it is written. Since the lower level name of “Adnus” is not used, “Baruch Shem” may be said aloud. This is also why Yakov Avinu said “Baruch Shem” aloud; he perceived Hash-m’s presence the way the angels do.
When we say the name of Hash-m in this world, however, we say only the name of “Adnus.” We praise that name by quietly intoning, “Baruch Shem.” The verse of “Baruch Shem” has two meanings. One is a lower form of praise of the name of “Adnus,” and one is a lofty form of praise of the name of “Yud-Heh.” Therefore, we say it quietly, like a person who has a message that can be understood in two ways, one that is lofty, and one that sounds incongruous. He whispers it so that the wise people who understand the lofty meaning will hear it and know that he is whispering it in order not to reveal the lofty wisdom behind it. The unlearned people will think that he is whispering it because it is a senseless statement and he is embarrassed to say it aloud. Accordingly, when the Gemara here implies that “Baruch Shem” is a lower form of praise, it does not contradict the other sources that imply that it is a higher form of praise. Both are correct, because both meanings are contained in “Baruch Shem.”