Per @RabbiKaii's request:
The reason that the teachings of the Zugot have been lost seems largely based on the reality of classic tannaitic and amoraic texts not including much of any of their teachings. Thus, while these works became more and more popular, other teachings were largely lost with time. Why these texts hardly included these teachings is a matter still heavily debated among academic scholars, though unless we uncover some lost works with clearer explanations (and I wouldn't lose hope for the possibility, every now and then we discover valuable lost works), or else wait for techiyat ha'metim, it seems like we'll never truly know for sure.
However, we do have a hint that a mishnaic compilation authored by personnel connected to the Chashmonaim or at least living in their time was still known circa the 4th-5th centuries. The Church Father Epiphanius, who was born a Jew in a small village near Beit Guvrin (Eleutheropolis), wrote in his book Panarion, I, 15:
"Scribes had four "repetitions." One in the name of the prophet Moses, a second in that of their teacher called Aqiba or Bar Aqiba, another in the name of Addan or Annan, also called Judas, and another in the name of the sons of Hasmonaeus. Whatever customs they derive from these four traditions under the impression that they are wisdom - they are unwisdom mostly - are boasted of and praised, and celebrated and acclaimed as the teaching to be given first place."
According to Prof. Y. N. Epstein in his book Introductions to the Tannaitic Literature, p. 17, the "recitations" are mishnaic compilations (the word mishna comes from שנה, recite) and Epiphanius is speaking about four mishnaic compilations revered by the Jews: The first is Sefer Devarim, or Mishneh Torah, authored by Moshe. The second is by Rabbi Akiva, the third is by Rebbi (Rabbi Yehudah Ha'Nasi, the other two names are mistaken versions of the variants יודה and יודן) and the last one is from "the sons of the Chasmonaim".
Prof. Epstein mentions two relevant sources that may be related to the Mishnaic compilation from the time of the Chashmonaim:
Two mentions of a halachic ruling of a "beit din of Chashmonai" (Avodah Zara 36b and Sanhedrin 82a), and a tradition brought in some versions of Halachot Gedolot which states that the elders of Beit Hillel and Beit Shammai wrote something called "the Scroll of the House of the Chashmonaim" (link, and see n. 9).
From all of these sources, it seems that traditions of teachings and rulings from at least some of the Zugot or their contemporaries disappeared at least a couple of centuries after the creation of Rebbi's Mishna.
As a side-note, I will add that a Kabbalistic work that was preserved in at least two places, one at the end of Brachot in the Munich 95 MS of the Bavli and another in a Kabbalistic compilation manuscript that had been owned for a time by my great-great-grandfather R' David Frankel and his partner and relative R' Lipa Schwager as part of a collection of rare books and manuscripts they sold in their bookshop in Husiatyn, Austria,1, mentions that part or all of this particular text had been transmitted thus:
"זה השם} מיכאל/מיכאל המלאך מסרו ליהודה ויהודה לבני חשמונאי}" ({This Name} Michael/Michael the Angel transmitted it to Yehudah and Yehudah to the Sons of Chashmonai).
1 This compilation, including quoting the relevant passage, is mentioned in one of Frankel & Schwager's catalogues. On Otzar Hachochmah it's called "רשימת ספרים ישנים גם חדשים - 40 - 7", p. 21:
I haven't not yet managed to discover whether this manuscript was preserved in some library or collection.
