The sixth Perek of Avos begins
שנו רבותינו בלשון המשנה ברוך שבחר בהם ובמשנותם
The Rabbis taught in the language of the Mishnah. Blessed is He Who chose them and their teachings.
Here I ask why this particular formulation is used, but the idea is that it makes the point that this Perek is not actually from Mishnayos Avos, but rather a Braisa (in this case, Kallah and Seder Eliyahu Zuta).
Why is it only here? Take the fourth Perek of Bikkurim as an example, Perek Adreigonus. It’s not actually from Mishnayos Bikkurim, but rather added from the second Perek of Tosefta Bikkurim. I don’t see that Perek beginning with this notation. For a single-Mishnah example, take Pesachim 4:9; the Gemara on 56a only quotes 4:8 and brings down 4:9 as ”Tanu Rabbanan” (which introduces a Braisa). So why is Avos 6 the only one that gets this notation?
While one could argue that it’s because it’s recited publicly, the words of Rashi on the Mishnah would seem to indicate otherwise:
שָׁנוּ חֲכָמִים בִּלְשׁוֹן הַמִּשְׁנָה. כְּלוֹמַר בְּרַיְתָא הִיא וּבִלְשׁוֹן הַמִּשְׁנָה הִיא שְׁנוּיָה, אֲבָל אֵינָהּ מִשְׁנָה. וּמַה שָּׁנוּ, רַבִּי מֵאִיר אוֹמֵר וְכוּ'. וּלְפִי שֶׁעַד עַכְשָׁיו כָּל הַפְּרָקִים מִשְׁנָה, לְפִיכָךְ הֻצְרַךְ לְהוֹדִיעַ שֶׁמִּכָּאן וְאֵילָךְ בְּרַיְתָא הִיא. וּמִתּוֹךְ שֶׁהַלָּלוּ דִּבְרֵי הַגָּדָה הֵן וּמְסַפְּרוֹת בְּעֵסֶק תַּלְמוּד תּוֹרָה נָהֲגוּ לְאָמְרָן בְּבֵית הַכְּנֶסֶת עִם שְׁאָר פְּרָקִים הַלָּלוּ שֶׁל מַסֶּכֶת אָבוֹת:
That is to say, this is a Braisa, taught in the language of the Mishnah, but it’s not a Mishnah. What was taught? “R’ Meir said, etc.” Since up until now all these Perakim were Mishnah, therefore it’s needed to inform that from here on is Braisa. Since these are words of Aggadah and discuss learning Torah, the Minhag is to say them in shul with the other perakim of Avos.
To me the wording implies “These are Braisos, therefore it needs to specify that these are not Mishnayos. Why was this included? Because we learn it in shul.” Not “We include this because we learn it in shul, therefore we need to tell people that these are Braisos” - seemingly anyone reading this would need to know they’re Braisos, independent of the reason this Perek is included.