I would like to offer my own novel explanation to this based on an idea which I have heard from Rav Shimshon Pincus ZTL.
Rav Pincus ZTL explains that when the 9th day of Av falls on shabbos, the reason why The mourning of Tiasha B'av is nidcheh, pushe off, is not only because we are forbidden to mourn on shabbos so we do it the next day instead, but for a much deeper reason.
There are quite few parellels between the Bais Hamikdash and Shabbos.
In the Sefer Nefesh Shimshon on Shabbos, (compiled posthumously from
the teachings of Rav Shimshon Pinkus), there is a whole section
enumerating the parallels between the holiness of Shabbos and the
holiness of the Bais HaMikdash. There is not room here to list all of them
but we can mention a few.
First of all he explains that the main purpose of the Mikdash is to
bring forgiveness to those who sinned. The Mikdash was called LEVANON
because SHEMALBIN AVONOTEI- HEN SHEL YISRAEL, it whitened (cleansed)
the sins of Israel (Yoma 39:). There was never anyone in Jerusalem who
had sins on him since the morning TAMID sacrifice brought forgiveness
for sins they did at night and the late afternoon Korban Tamid sacrifice for
sins done during the day. The Talmud says that Shabbat had the same
capacity. “Anyone who says Vaychulu Hashamayim… (which we say in
Kiddush) all his sins are forgiven” (Shabbos 119:). Anyone who
realizes that Hashem is the one who created the world (as we state in
Kiddush) and who runs it even today realizes that he can’t do anything
against Hashem and he will not sin.
All the malachos – creative work that we are forbidden to do on
Shabbat is learned from the work done in building the Mishkan.
Special clothing for Shabbos, like the special clothing of the Kohanim
serving in the Mishkan / Bais Hamikdash.
In the Mikdash there were Lechem Hapanim, On Shabbos we hve special
breads, challos, and we put specific emphasis on the bread by doubling
them with lechm mishnah.
Shabbos has special songs. The Zemiros of Shabbos. Each day a
specific psalm was sung in the Bais hamikdash. As added emphasis, the
song for shabbos is the only one which the passuk itself prefaces with
the opening words "Mizmor Shir L'Yom Hashabbos".
He even goes on to connect the special smells of Shabbos,with the
smells of the Ketores i the bais hamikdash. See quote and story in
this article.
Based on these points, Rav Pincus explained the idea that the building
of the Bais Hamikdash was interrupted for shabbos. See Gemara Yevamos
6a. (He added that although the Gemara there gives an explanation
of כולכם חייבין בכבודי, there is deep understanding to this idea).
He explains that the shabbos is somewhat of an alternative, (and
somewhat higher) manifestation of the concept of Bais Hamikdash.
Therefore, We interrupt it's construction for Shabbos.
With this idea he explains the postpoement of the mourning of the
destruction of the Bais Hamikdash when tisha B'Av falls on shabbos.
"When we have Shabbos, we have Bais Hamikdash", we therefore do not
mourn it's dstruction on shabbos.
__ ____ ____
Based on this concept, IMHO, we can understand why Rebbe posited that "הואיל ונדחה ידחה" - The year which Tisha B'Av fell on shabbos "we had Bais Hamikdash on Tisha B'av", and thus there is no need to reeinstate mourning for the Bais Hamikdash anymore that year.
(the Chachamim, who disagreed with Rebbe, opined that this concept is enough to postpone the mourning from shabbos, but not to do away with it completely.
This can also answer the (poster's second) question, why does [Rebbe] only argue on Tishah b’Av? Several fast days can fall out on Shabbos...?
If this writer's explanation is correct, it only applies to Tisha B'av but not to other fasts of which the reason for the postponement is only due to the prohibition of fasting on shabbos.
