You are asking about midrachim reported regarding the special halachic status of BSKMLV and additionally you are asking about people who are reciting it in an audible way everyday.
First, the midrashim
I seem to recall reading somewhere that the authorities in ancient Babylon heard Baruch Shem being recited in a synagogue and interpreted this to be an breach of alligience to the king. This then resulted in hostility towards the Jewish community, prompting the Rabbis to prohibit reciting Baruch Shem out loud, except on Yom Kippur. I don't know this, anyway in halacha, based on Gemara this is not the reason.
See Shulchan Aruch Orach Chayim siman 61, 13, Mishna Berura sk 30 which quotes the Gemara Pesachim 56a. I will quote a more extended stuff. :
And what is the reason that we do recite it? - Even as Rabbi Simeon Ben Lakish expounded. For Rabbi Simeon Ben Lakish said: And Jacob called unto his sons, and said: Gather yourselves together, that I may tell you [that which shall befall you in the end of days].
Jacob wished to reveal to his sons the 'end of the days', whereupon the Shechinah departed from him.
Said he, 'Perhaps, Heaven forfend! there is one unfit among my children, like Abraham, from whom there issued Ishmael, or like my father Isaac, from whom there issued Esau.'
[But] his sons answered him, 'Hear O Israel, the Lord our God the Lord is One: just as there is only One in thy heart, so is there in our heart only One.'
In that moment our father Jacob opened [his mouth] and exclaimed, 'Blessed be the name of His glorious kingdom for ever and ever.'
Said the Rabbis, How shall we act?
Shall we recite it, - but our Teacher Moses did not say it.
Shall we not say it - but Jacob said it! [Hence] they enacted that it should be recited quietly(¹).
Rabbi Isaac said, The School of Rabbi Ammi said: This is to be compared to a king's daughter who smelled a spicy pudding (Rashi, meat spiced).
If she reveals [her desire], she suffers disgrace; if she does not reveal it, she suffers pain.
So her servants began bringing it to her in secret.
Rabbi Abbahu(²) said: They [the Sages] enacted that this should be recited aloud, on account of the resentment of heretics (who did not say BSKMLV).
But in Nehardea, where there are no heretics so far, they recite it quietly.
(¹) This is a compromise as you say partially out loud, seems not linked to Chabbad. I could speculate that since some of them believe that we are already living in Messianic times, therefore the prohibition against saying Baruch Shem out loud is no longer valid. I think that this speculation is not right.
(²) There was a period in which it was important to inform people about the importance to say it.
For Yom Kippur see SA OC 619, 2, Magen Avraham sk 8.
He says that all the year, we say BSKMLV silently because Moshe did steal it from angels, and in Yom Kippur we are ourselves as angels, so it's our, we need no more hide it. The source of this explanation is in Tur in name of the Midrash Rabba on Psrashat Vaetchanan. The Midrash says that Moshe heard this praise of G-d by angels in heaven and taught it to Israel. An allegory is: Someone did steal a jewel in the king's palace, and gave it to his wife. He said her to wear it at home only.
Note: The two explanations, the first regarding everyday and the second regarding Yom Kippurim seem to be contradictory, but both are reported by poskim. Their respective meanings are perhaps not contradictory at all, but I am not able to understand this now.
secondly
the fact that we say BSKMLV quietly does not mean that it needs to be inaudible as noted @Yaakov Deane. The Mishna says that the reader needs to be heard by his ears.
