Whatsaktome,
The thing is you’re missing the limud and context behind the statement that the “(previous) Rebbe will lead us out of exile”. If you look at the source of that statement, in Bosi L’gani yud alef, the Rebbe put a footnote corresponding to that statement to explain what was meant by it. I assume you haven’t learned it seeing as you are learning out the wrong context for the statement.
The context the Rebbe provides for the statement in the footnote is to perek yud tes (if I’m remembering correctly) of Bamidbor Rabba, and to the end of Sefer Chassidim, in both those places there’s expressed a similar idea: In Vayikra Rabba, it details a conversation between HKB”H and Moshe Rabbeinu, in which HKB”H responds to Moshe Rabbeinu’s request to go into eretz Yisroel:
אָמַר לוֹ הַקָּדוֹשׁ בָּרוּךְ הוּא לְמשֶׁה בְּאֵיזֶה פָּנִים אַתָּה מְבַקֵּשׁ לִכָּנֵס לָאָרֶץ, מָשָׁל לְרוֹעֶה שֶׁיָּצָא לִרְעוֹת צֹאנוֹ שֶׁל מֶלֶךְ וְנִשְׁבֵּית הַצֹּאן, בִּקֵּשׁ הָרוֹעֶה לִכָּנֵס לַפַּלְטֵרִין שֶׁל מֶלֶךְ, אָמַר לוֹ הַמֶּלֶךְ אִם אַתְּ נִכְנַס עַכְשָׁו מַה יֹּאמְרוּ הַבְּרִיּוֹת שֶׁאַתָּה הִשְׁבֵּיתָ הַצֹּאן. אַף כָּאן אָמַר לוֹ הַקָּדוֹשׁ בָּרוּךְ הוּא לְמשֶׁה, שִׁבְחֲךָ הוּא שֶׁהוֹצֵאתָ שִׁשִּׁים רִבּוֹא וּקְבַרְתָּם בַּמִּדְבָּר, וְאַתְּ מַכְנִיס דּוֹר אַחֵר, עַכְשָׁיו יֹאמְרוּ אֵין לְדוֹר הַמִּדְבָּר חֵלֶק לָעוֹלָם הַבָּא, אֶלָּא תְּהֵא בְּצִדָּן וְתָבוֹא עִמָּהֶן, שֶׁנֶּאֱמַר (דברים לג, כא): וַיֵּתֵא רָאשֵׁי עָם צִדְקַת ה’ עָשָׂה, לְכָךְ כְּתִיב (במדבר כ, יב): לֹא תָבִיאוּ אֶת הַקָּהָל הַזֶּה, אֶלָּא שֶׁיָּצָא עִמָּךְ.
“You will not bring this group (into the land which I have given you), rather it will enter with you.” That when moshiach comes (may it be speedily), Moshe Rabbeinu will lead his generation to greet moshiach (this is also the concept found at the end of Sefer HaChassidim). That does NOT imply that Moshe Rabbeinu will be moshiach, rather that he will help his generation meet moshiach and that they will be redeemed in his merit. That is what the Rebbe is saying with the statement you quoted, and THAT is why the Rebbe brings a minority opinion about techiya before moshiach, to explain how possibly the Rebbe RaYa”TZ ZY”A could lead his generation to greet moshiach.
This is also why the Rebbe ZY”A was constantly ADAMANT that people learn his footnotes! Because without doing so one can come to a completely mistaken understanding of the text as you did above.