I've learned that the basic transmission of the oral law was: Moses > Joshua > Elders > Prophets > Great Assembly > Specific Rabbis
But this seems to run contradictory from what we'd expect from the Written Torah, which states that the Levites and Priests would be the ones teaching the people the laws and statutes of the Torah.
וַיְדַבֵּ֣ר יְהֹוָ֔ה אֶֽל־אַהֲרֹ֖ן לֵאמֹֽר׃
יַ֣יִן וְשֵׁכָ֞ר אַל־תֵּ֣שְׁתְּ ׀ אַתָּ֣ה ׀ וּבָנֶ֣יךָ אִתָּ֗ךְ בְּבֹאֲכֶ֛ם אֶל־אֹ֥הֶל מוֹעֵ֖ד וְלֹ֣א תָמֻ֑תוּ חֻקַּ֥ת עוֹלָ֖ם לְדֹרֹתֵיכֶֽם׃ וּֽלְהַבְדִּ֔יל בֵּ֥ין הַקֹּ֖דֶשׁ וּבֵ֣ין הַחֹ֑ל וּבֵ֥ין הַטָּמֵ֖א וּבֵ֥ין הַטָּהֽוֹר׃
And יהוה spoke to Aaron, saying:
Drink no wine or other intoxicant, you or your sons, when you enter the Tent of Meeting, that you may not die. This is a law for all time throughout the ages,
for you must distinguish between the sacred and the profane, and between the impure and the pure; and you must teach the Israelites all the laws which יהוה has imparted to them through Moses.
וּלְלֵוִ֣י אָמַ֔ר תֻּמֶּ֥יךָ וְאוּרֶ֖יךָ לְאִ֣ישׁ חֲסִידֶ֑ךָ אֲשֶׁ֤ר נִסִּיתוֹ֙ בְּמַסָּ֔ה תְּרִיבֵ֖הוּ עַל־מֵ֥י מְרִיבָֽה׃
הָאֹמֵ֞ר לְאָבִ֤יו וּלְאִמּוֹ֙ לֹ֣א רְאִיתִ֔יו וְאֶת־אֶחָיו֙ לֹ֣א הִכִּ֔יר וְאֶת־בָּנָ֖ו לֹ֣א יָדָ֑ע כִּ֤י שָֽׁמְרוּ֙ אִמְרָתֶ֔ךָ וּבְרִיתְךָ֖ יִנְצֹֽרוּ׃ יוֹר֤וּ מִשְׁפָּטֶ֙יךָ֙ לְיַעֲקֹ֔ב וְתוֹרָתְךָ֖ לְיִשְׂרָאֵ֑ל יָשִׂ֤ימוּ קְטוֹרָה֙ בְּאַפֶּ֔ךָ וְכָלִ֖יל עַֽל־מִזְבְּחֶֽךָ׃
And of Levi he said: Let Your Thummim and Urim Be with Your faithful one, Whom You tested at Massah, Challenged at the waters of Meribah;
Who said of his father and mother, “I consider them not.” His brothers he disregarded, Ignored his own children. Your precepts alone they observed, And kept Your covenant. They shall teach Your laws to Jacob And Your instructions to Israel. They shall offer You incense to savor* And whole-offerings on Your altar.
כִּֽי־שִׂפְתֵ֤י כֹהֵן֙ יִשְׁמְרוּ־דַ֔עַת וְתוֹרָ֖ה יְבַקְשׁ֣וּ מִפִּ֑יהוּ כִּ֛י מַלְאַ֥ךְ יְהֹֽוָה־צְבָא֖וֹת הֽוּא׃ וְאַתֶּם֙ סַרְתֶּ֣ם מִן־הַדֶּ֔רֶךְ הִכְשַׁלְתֶּ֥ם רַבִּ֖ים בַּתּוֹרָ֑ה שִֽׁחַתֶּם֙ בְּרִ֣ית הַלֵּוִ֔י אָמַ֖ר יְהֹוָ֥ה צְבָאֽוֹת׃
For the lips of a priest guard knowledge, And men seek rulings from his mouth; For he is a messenger of the LORD of Hosts. But you have turned away from that course: You have made the many stumble through your rulings;d you have corrupted the covenant of the Levites—said the LORD of Hosts.
From the first book (the Torah) to the last prophetic book (Malakhi) it's taken for granted that the teaching of the law and its statutes was done by the priests, and that it was the responsibility of the priests. If this is the case, why do most of the "Oral Law" chains seem to sidestep most levites and priests?
Is it possible this "sidestep" in the chain has to do with the reality that most priests were teaching that there was no oral law during the times of the Mishnah?