Of the three sources you quoted, I think that all MY readers would benefit from seeing the context of the quotes, which will help answer the question.
For the first source (the previous Lubavitcher Rebbe), he continues by quoting the Passuk in Mishlei 14:1:
חַכְמוֹת נָשִׁים בָּנְתָה בֵיתָהּ וְאִוֶּלֶת בְּיָדֶיהָ תֶהֶרְסֶנּוּ
Every wise woman buildeth her house; But the foolish plucketh it down with her hands. (JPS)
(in his words "וכתי' חכמות כו' בנתה ביתה עפ"י התורה והמצוה.")
Perhaps this is his source right here, that it is the woman's job to build the Bayis, as in, keep the Shalom Bayis, which works especially well with Hashem having given women "Binah Yeseirah" (see here also). This is slightly unlikely, however, as he seems to be adding another point by bringing this Passuk.
The second source, the Lubavitcher Rebbe is clearly speaking from practical experience, as evidenced by his wording and the context of his statement in a letter. Please read the letter in the link, where he is offering advice to a couple to help resolve some Shalom Bayis issue.
The third source, Rambam Ishus 15:20, is most clear. He says:
וכן צוו על האשה שתהיה מכבדת את בעלה ביותר מדאי ויהיה עליה מורא ממנו ותעשה כל מעשיה על פיו. ויהיה בעיניה כמו שר או מלך מהלכת בתאות לבו ומרחקת כל מה שישנא. וזה דרך בנות ישראל ובני ישראל הקדושים והטהורים בזיווגן. ובדרכים אלו יהיה ישובן נאה ומשובח:
And similarly, they commanded a woman to honor her husband exceedingly and to be in awe of him. She should carry out all her deeds according to his directives, considering him to be an officer or a king. She should follow the desires of his heart and shun everything that he disdains. This is the custom of holy and pure Jewish women and men in their marriages. And these ways will make their marriage pleasant and praiseworthy.
Rambam (after listing the requirements of the man to his wife, see Halacha 19 there) lists the requirements of the woman towards her husband, including how she should listen to him etc. etc. is clearly focusing on the wife, and as a result, she is listed first.
This would also serve as a source for "the Torah putting the burden of Shalom Bayis on the wife". Rambam certainly sounds like that when you read his words above (see the bolded terms, where Rambam assumes it is her responsibility to "follow the desires of his heart, and shun everything he disdains).
Feedback encouraged as usual, but I feel like this is at the very least an adequate answer to the question.