The Jewish Encyclopedia mentions Mishnah's failure to mention Gamaliel. Could be a challenge to this understanding - unless Christianity was mythically intimated to be a second branch of Judaism. But look again at Abot 1. An argument could be made for a continuation from Gamaliel to R. Judah ha Nasi:
1
משֶׁה קִבֵּל תּוֹרָה מִסִּינַי, וּמְסָרָהּ לִיהוֹשֻׁעַ, וִיהוֹשֻׁעַ לִזְקֵנִים, וּזְקֵנִים לִנְבִיאִים, וּנְבִיאִים מְסָרוּהָ לְאַנְשֵׁי כְנֶסֶת הַגְּדוֹלָה. הֵם אָמְרוּ שְׁלשָׁה דְבָרִים, הֱווּ מְתוּנִים בַּדִּין, וְהַעֲמִידוּ תַלְמִידִים הַרְבֵּה, וַעֲשׂוּ סְיָג לַתּוֹרָה:
Moshe received the Torah from Sinai and transmitted it to Yehoshua, and Yehoshua to the Elders, and the Elders to the Prophets, and the Prophets transmitted it to the Men of the Great Assembly. They said three things: Be deliberate in judgment, raise up many disciples and make a fence for the Torah.
2
שִׁמְעוֹן הַצַּדִּיק הָיָה מִשְּׁיָרֵי כְנֶסֶת הַגְּדוֹלָה. הוּא הָיָה אוֹמֵר, עַל שְׁלשָׁה דְבָרִים הָעוֹלָם עוֹמֵד, עַל הַתּוֹרָה וְעַל הָעֲבוֹדָה וְעַל גְּמִילוּת חֲסָדִים:
Shimon the Righteous was from the remnants of the Great Assembly. He would say, "On three things the world stands: on the Torah, on the service and on acts of lovingkindness."
3
אַנְטִיגְנוֹס אִישׁ סוֹכוֹ קִבֵּל מִשִּׁמְעוֹן הַצַּדִּיק. הוּא הָיָה אוֹמֵר, אַל תִּהְיוּ כַעֲבָדִים הַמְשַׁמְּשִׁין אֶת הָרַב עַל מְנָת לְקַבֵּל פְּרָס, אֶלָּא הֱווּ כַעֲבָדִים הַמְשַׁמְּשִׁין אֶת הָרַב שֶׁלֹּא עַל מְנָת לְקַבֵּל פְּרָס, וִיהִי מוֹרָא שָׁמַיִם עֲלֵיכֶם:
Antigonos, man of Sokho, received from Shimon the Righteous. He would say, "Do not be as servants who are serving the master in order to receive a reward, rather be as servants who are serving the master not in order to receive a reward; and may the fear of Heaven be upon you."
4
יוֹסֵי בֶן יוֹעֶזֶר אִישׁ צְרֵדָה וְיוֹסֵי בֶן יוֹחָנָן אִישׁ יְרוּשָׁלַיִם קִבְּלוּ מֵהֶם. יוֹסֵי בֶן יוֹעֶזֶר אִישׁ צְרֵדָה אוֹמֵר, יְהִי בֵיתְךָ בֵית וַעַד לַחֲכָמִים, וֶהֱוֵי מִתְאַבֵּק בַּעֲפַר רַגְלֵיהֶם, וֶהֱוֵי שׁוֹתֶה בְצָמָא אֶת דִּבְרֵיהֶם:
Yose ben Yoezer, man of Tsreida, and Yose ben Yochanan, man of Jerusalem, received from him. Yose ben Yoezer says, "May your house be a meeting house for Sages, become dirty in the dust of their feet and drink their words thirstily."
5
יוֹסֵי בֶן יוֹחָנָן אִישׁ יְרוּשָׁלַיִם אוֹמֵר, יְהִי בֵיתְךָ פָתוּחַ לִרְוָחָה, וְיִהְיוּ עֲנִיִּים בְּנֵי בֵיתֶךָ, וְאַל תַּרְבֶּה שִׂיחָה עִם הָאִשָּׁה. בְּאִשְׁתּוֹ אָמְרוּ, קַל וָחֹמֶר בְּאֵשֶׁת חֲבֵרוֹ. מִכָּאן אָמְרוּ חֲכָמִים, כָּל זְמַן שֶׁאָדָם מַרְבֶּה שִׂיחָה עִם הָאִשָּׁה, גּוֹרֵם רָעָה לְעַצְמוֹ, וּבוֹטֵל מִדִּבְרֵי תוֹרָה, וְסוֹפוֹ יוֹרֵשׁ גֵּיהִנֹּם:
Yose ben Yochanan, man of Jerusalem, says, "May your home be open wide, may the poor be members of your household and do not increase conversation with the woman." They so stated with his wife; all the more so with the wife of his friend. From this, the sages said, "Any time that a man increases conversation with the woman, he causes evil to himself and neglects the words of Torah; and, in his end, he inherits Geihinam."
6
יְהוֹשֻׁעַ בֶּן פְּרַחְיָה וְנִתַּאי הָאַרְבֵּלִי קִבְּלוּ מֵהֶם. יְהוֹשֻׁעַ בֶּן פְּרַחְיָה אוֹמֵר, עֲשֵׂה לְךָ רַב, וּקְנֵה לְךָ חָבֵר, וֶהֱוֵי דָן אֶת כָּל הָאָדָם לְכַף זְכוּת:
Yehoshua ben Perachiah and Nitai of Arbel received from them. Yehoshua ben Perachia says, "Make for yourself a mentor, acquire for yourself a friend and judge every person as meritorious."
7
נִתַּאי הָאַרְבֵּלִי אוֹמֵר, הַרְחֵק מִשָּׁכֵן רָע, וְאַל תִּתְחַבֵּר לָרָשָׁע, וְאַל תִּתְיָאֵשׁ מִן הַפֻּרְעָנוּת:
Nitai of Arbel says: "Distance [yourself] from a bad neighbor, do not befriend an evildoer and do not despair of punishment."
8
יְהוּדָה בֶן טַבַּאי וְשִׁמְעוֹן בֶּן שָׁטָח קִבְּלוּ מֵהֶם. יְהוּדָה בֶן טַבַּאי אוֹמֵר, אַל תַּעַשׂ עַצְמְךָ כְעוֹרְכֵי הַדַּיָּנִין. וּכְשֶׁיִּהְיוּ בַעֲלֵי דִינִין עוֹמְדִים לְפָנֶיךָ, יִהְיוּ בְעֵינֶיךָ כִרְשָׁעִים. וּכְשֶׁנִּפְטָרִים מִלְּפָנֶיךָ, יִהְיוּ בְעֵינֶיךָ כְזַכָּאִין, כְּשֶׁקִּבְּלוּ עֲלֵיהֶם אֶת הַדִּין:
Yehuda ben Tabai and Shimon ben Shetach received from them. Yehuda ben Tabai says, "Do not make yourself like the judges' advisers; and when the litigants are before you, they should be like evildoers in your eyes; and when they are excused from before you, they should be meritorious in your eyes - when they have accepted the judgment."
9
שִׁמְעוֹן בֶּן שָׁטָח אוֹמֵר, הֱוֵי מַרְבֶּה לַחְקֹר אֶת הָעֵדִים, וֶהֱוֵי זָהִיר בִּדְבָרֶיךָ, שֶׁמָּא מִתּוֹכָם יִלְמְדוּ לְשַׁקֵּר:
Shimon ben Shatach says, "Examine the witnesses thoroughly, but be careful with your words, lest from them they learn to lie."
10
שְׁמַעְיָה וְאַבְטַלְיוֹן קִבְּלוּ מֵהֶם. שְׁמַעְיָה אוֹמֵר, אֱהֹב אֶת הַמְּלָאכָה, וּשְׂנָא אֶת הָרַבָּנוּת, וְאַל תִּתְוַדַּע לָרָשׁוּת:
Shemayah and Avtalyon received from them. Shemayah says, "Love work, hate lordship and do not become familiar with the government."
11
אַבְטַלְיוֹן אוֹמֵר, חֲכָמִים, הִזָּהֲרוּ בְדִבְרֵיכֶם, שֶׁמָּא תָחוּבוּ חוֹבַת גָּלוּת וְתִגְלוּ לִמְקוֹם מַיִם הָרָעִים, וְיִשְׁתּוּ הַתַּלְמִידִים הַבָּאִים אַחֲרֵיכֶם וְיָמוּתוּ, וְנִמְצָא שֵׁם שָׁמַיִם מִתְחַלֵּל:
Avtalyon says, "Sages, be careful with your words, lest you become obligated in an obligation of exile and are exiled to the place of evil waters, and the students who follow after you will drink, and thus the name of Heaven is profaned."
12
הִלֵּל וְשַׁמַּאי קִבְּלוּ מֵהֶם. הִלֵּל אוֹמֵר, הֱוֵי מִתַּלְמִידָיו שֶׁל אַהֲרֹן, אוֹהֵב שָׁלוֹם וְרוֹדֵף שָׁלוֹם, אוֹהֵב אֶת הַבְּרִיּוֹת וּמְקָרְבָן לַתּוֹרָה:
Hillel and Shammai received from them. Hillel says, "Be of the disciples of Aharon, loving peace and pursuing peace, loving the creatures and bringing them closer to Torah."
13
הוּא הָיָה אוֹמֵר, נָגֵד שְׁמָא, אָבֵד שְׁמֵהּ. וּדְלֹא מוֹסִיף, יָסֵף. וּדְלֹא יָלֵיף, קְטָלָא חַיָּב. וּדְאִשְׁתַּמֵּשׁ בְּתָגָא, חָלֵף:
He would say, "Spread a name, lose his name. And one who does not increase [knowledge] ceases. And one who does not study is liable to die. And one who makes use of the crown [of learning] passes away."
14
הוּא הָיָה אוֹמֵר, אִם אֵין אֲנִי לִי, מִי לִי. וּכְשֶׁאֲנִי לְעַצְמִי, מָה אֲנִי. וְאִם לֹא עַכְשָׁיו, אֵימָתָי:
He [Rabbi Hillel] used to say: If I am not for me, who will be for me? And when I am for myself alone, what am I? And if not now, then when?
15
שַׁמַּאי אוֹמֵר, עֲשֵׂה תוֹרָתְךָ קֶבַע. אֱמֹר מְעַט וַעֲשֵׂה הַרְבֵּה, וֶהֱוֵי מְקַבֵּל אֶת כָּל הָאָדָם בְּסֵבֶר פָּנִים יָפוֹת:
Shammai says, "Make your Torah fixed, say little and do much, and receive every person with a pleasant countenance."
16
רַבָּן גַּמְלִיאֵל הָיָה אוֹמֵר, עֲשֵׂה לְךָ רַב, וְהִסְתַּלֵּק מִן הַסָּפֵק, וְאַל תַּרְבֶּה לְעַשֵּׂר אֹמָדוֹת:
Rabban Gamliel says, "Make for yourself a mentor, remove yourself from doubt and do not frequently tithe by estimation."
17
שִׁמְעוֹן בְּנוֹ אוֹמֵר, כָּל יָמַי גָּדַלְתִּי בֵין הַחֲכָמִים, וְלֹא מָצָאתִי לַגּוּף טוֹב אֶלָּא שְׁתִיקָה. וְלֹא הַמִּדְרָשׁ הוּא הָעִקָּר, אֶלָּא הַמַּעֲשֶׂה. וְכָל הַמַּרְבֶּה דְבָרִים, מֵבִיא חֵטְא:
Shimon, his son, says, "All my days I grew up among the Sages, and I did not find anything good for the body except silence. And the exposition [of Torah] is not what is essential, but the action. And whoever increases words brings sin."
18
רַבָּן שִׁמְעוֹן בֶּן גַּמְלִיאֵל אוֹמֵר, עַל שְׁלשָׁה דְבָרִים הָעוֹלָם עוֹמֵד, עַל הַדִּין וְעַל הָאֱמֶת וְעַל הַשָּׁלוֹם, שֶׁנֶּאֱמַר (זכריה ח) אֱמֶת וּמִשְׁפַּט שָׁלוֹם שִׁפְטוּ בְּשַׁעֲרֵיכֶם:
Rabban Shimon ben Gamliel says, "On three things the world stands: on judgment, on truth and on peace, as it is said (Zachariah 8:16), 'Judge truth and the justice of peace in your gates.'"
2
1
רַבִּי אוֹמֵר, אֵיזוֹהִי דֶרֶךְ יְשָׁרָה שֶׁיָּבֹר לוֹ הָאָדָם, כֹּל שֶׁהִיא תִפְאֶרֶת לְעוֹשֶׂיהָ וְתִפְאֶרֶת לוֹ מִן הָאָדָם. וֶהֱוֵי זָהִיר בְּמִצְוָה קַלָּה כְבַחֲמוּרָה, שֶׁאֵין אַתָּה יוֹדֵעַ מַתַּן שְׂכָרָן שֶׁל מִצְוֹת. וֶהֱוֵי מְחַשֵּׁב הֶפְסֵד מִצְוָה כְּנֶגֶד שְׂכָרָהּ, וּשְׂכַר עֲבֵרָה כְנֶגֶד הֶפְסֵדָהּ. וְהִסְתַּכֵּל בִּשְׁלשָׁה דְבָרִים וְאִי אַתָּה בָא לִידֵי עֲבֵרָה, דַּע מַה לְּמַעְלָה מִמְּךָ, עַיִן רוֹאָה וְאֹזֶן שׁוֹמַעַת, וְכָל מַעֲשֶׂיךָ בַסֵּפֶר נִכְתָּבִין:
Rabbi [Yehuda haNasi] said: Which is the straight path that a person should choose for himself? Whichever [path] that is [itself] praiseworthy for the person adopting [it], And praiseworthy to him from [other] people. And be as careful with a light commandment as with a weighty one, for you do not know the reward given [for the fulfillment] of [the respective] commandments. Also, weigh the loss [that may be sustained through the fulfillment] of a commandment against the reward [that may be obtained] for [fulfilling] it. And [weigh] the gain [that may be obtained through the committing] of a transgression against the loss [that may be sustained] by [committing] it. Keep your eye on three things, and you will not come to sin: Know what is above you: An Eye that sees, and an Ear that hears, and all your deeds are written in a book.
2
רַבָּן גַּמְלִיאֵל בְּנוֹ שֶׁל רַבִּי יְהוּדָה הַנָּשִׂיא אוֹמֵר, יָפֶה תַלְמוּד תּוֹרָה עִם דֶּרֶךְ אֶרֶץ, שֶׁיְּגִיעַת שְׁנֵיהֶם מְשַׁכַּחַת עָוֹן. וְכָל תּוֹרָה שֶׁאֵין עִמָּהּ מְלָאכָה, סוֹפָהּ בְּטֵלָה וְגוֹרֶרֶת עָוֹן. וְכָל הָעֲמֵלִים עִם הַצִּבּוּר, יִהְיוּ עֲמֵלִים עִמָּהֶם לְשֵׁם שָׁמַיִם, שֶׁזְּכוּת אֲבוֹתָם מְסַיַּעְתָּן וְצִדְקָתָם עוֹמֶדֶת לָעַד. וְאַתֶּם, מַעֲלֶה אֲנִי עֲלֵיכֶם שָׂכָר הַרְבֵּה כְּאִלּוּ עֲשִׂיתֶם:
Rabban Gamliel the son of Rabbi Yehudah HaNasi said: Excellent is the study of the Torah together with a worldly occupation; for the exertion [expended] in both of them causes sin to be forgotten. And all [study of the] Torah in the absence of a worldly occupation comes to nothing in the end and leads to sin. And all who work for the community, let them work for the [sake of the] name of Heaven; for the merit of their ancestors sustains them, And their righteousness (tsidkatam) will endure forever. And as for you [who work for the community, God says:] I credit you with a great reward, as if you [yourselves] had done it [on your own].
3
הֱווּ זְהִירִין בָּרָשׁוּת, שֶׁאֵין מְקָרְבִין לוֹ לָאָדָם אֶלָּא לְצֹרֶךְ עַצְמָן. נִרְאִין כְּאוֹהֲבִין בִּשְׁעַת הֲנָאָתָן, וְאֵין עוֹמְדִין לוֹ לָאָדָם בִּשְׁעַת דָּחְקוֹ:
Be careful about the government, as they approach a man only when they need him. They seem like good friends in good times, but they don't stay for him in time of his trouble.
4
הוּא הָיָה אוֹמֵר, עֲשֵׂה רְצוֹנוֹ כִרְצוֹנְךָ, כְּדֵי שֶׁיַּעֲשֶׂה רְצוֹנְךָ כִרְצוֹנוֹ. בַּטֵּל רְצוֹנְךָ מִפְּנֵי רְצוֹנוֹ, כְּדֵי שֶׁיְּבַטֵּל רְצוֹן אֲחֵרִים מִפְּנֵי רְצוֹנֶךָ. הִלֵּל אוֹמֵר, אַל תִּפְרֹשׁ מִן הַצִּבּוּר, וְאַל תַּאֲמִין בְּעַצְמְךָ עַד יוֹם מוֹתְךָ, וְאַל תָּדִין אֶת חֲבֵרְךָ עַד שֶׁתַּגִּיעַ לִמְקוֹמוֹ, וְאַל תֹּאמַר דָּבָר שֶׁאִי אֶפְשָׁר לִשְׁמֹעַ, שֶׁסּוֹפוֹ לְהִשָּׁמַע. וְאַל תֹּאמַר לִכְשֶׁאִפָּנֶה אֶשְׁנֶה, שֶׁמָּא לֹא תִפָּנֶה:
He was accustomed to say: Make His [God's] will like your will, so that He will make your will like His will. Nullify your will to His will, so that He will nullify the will of others to your will. Hillel says: Do not separate yourself from the community. Do not believe in yourself until the day of your death. Do not judge your fellow until you come to his place. Do not say something that cannot be heard, for in the end it will be heard. Do not say, "When I will be available I will study [Torah]," lest you never become available.
5
הוּא הָיָה אוֹמֵר, אֵין בּוּר יְרֵא חֵטְא, וְלֹא עַם הָאָרֶץ חָסִיד, וְלֹא הַבַּיְשָׁן לָמֵד, וְלֹא הַקַּפְּדָן מְלַמֵּד, וְלֹא כָל הַמַּרְבֶּה בִסְחוֹרָה מַחְכִּים. וּבְמָקוֹם שֶׁאֵין אֲנָשִׁים, הִשְׁתַּדֵּל לִהְיוֹת אִישׁ:
He was accustomed to say: A boor cannot fear sin. An ignorant person cannot be pious. A person prone to being ashamed cannot learn. An impatient person cannot teach. Not all who engage in a lot of business become wise. In a place where there is no man, strive to be a man.
6
אַף הוּא רָאָה גֻלְגֹּלֶת אַחַת שֶׁצָּפָה עַל פְּנֵי הַמַּיִם. אָמַר לָהּ, עַל דַּאֲטֵפְתְּ, אַטְפוּךְ. וְסוֹף מְטִיפַיִךְ יְטוּפוּן:
He also saw a skull that was floating on top of the water. He said (to it): "Since you drowned [others, others] drowned you. And in the end, those that drowned you will be drowned.
7
הוּא הָיָה אוֹמֵר, מַרְבֶּה בָשָׂר, מַרְבֶּה רִמָּה. מַרְבֶּה נְכָסִים, מַרְבֶּה דְאָגָה. מַרְבֶּה נָשִׁים, מַרְבֶּה כְשָׁפִים. מַרְבֶּה שְׁפָחוֹת, מַרְבֶּה זִמָּה. מַרְבֶּה עֲבָדִים, מַרְבֶּה גָזֵל. מַרְבֶּה תוֹרָה, מַרְבֶּה חַיִּים. מַרְבֶּה יְשִׁיבָה, מַרְבֶּה חָכְמָה. מַרְבֶּה עֵצָה, מַרְבֶּה תְבוּנָה. מַרְבֶּה צְדָקָה, מַרְבֶּה שָׁלוֹם. קָנָה שֵׁם טוֹב, קָנָה לְעַצְמוֹ. קָנָה לוֹ דִבְרֵי תוֹרָה, קָנָה לוֹ חַיֵּי הָעוֹלָם הַבָּא:
He was accustomed to say: The more flesh, the more worms. The more possessions, the more worry. The more wives, the more witchcraft. The more maidservants, the more lewdness. The more man-servants, the more theft. The more Torah, the more life. The more sitting [and studying], the more wisdom. The more counsel, the more understanding. The more charity, the more peace. One who has acquired a good name has acquired for himself. One who has acquired words of Torah has acquired for himself the life of the World to Come.
I think that's a pretty strong case for Gamaliel being R Judah's claim to tradition.