Tikkun Olam is a great idea. It's just not in Aleinu.

Regarding the correct spelling of le-taken olam in the Aleinu prayer, mistakenly translated in this context as "to repair the world," Mitchell First has written convincingly on this topic. The earliest textual evidence certainly points to לתכן ("to establish") instead of the prevalent לתקן ("to repair").

I would add, however, that the strongest proof for לתכן is contextual rather than from textual variants -- that is, from the Musaf prayer itself and from related biblical verses from which Aleinu's author seems to draw.

The expression לתכן עולם appears to allude to the second Ketuvim verse (Ps. 93:1) in the Malkhuyot section of Musaf: 

ה' מָלָךְ גֵּאוּת לָבֵשׁ לָבֵשׁ יְהֹוָה עֹז הִתְאַזָּר אַף־תִּכּוֹן תֵּבֵל בַּל־תִּמּוֹט

More precisely, לתכן עולם במלכות ש-די is a compound phrase constructed from the last words of that verse -- לתכן עולם follows תִּכּוֹן תֵּבֵל -- combined with the biblical usage תכון מלכות (e.g., I Sam. 20:31; I Kings 2:12), the establishment of a permanent dynasty or kingdom. 

But no matter how you spell the word in Aleinu, the current usage of Tikkun Olam represents a noble effort that, in manageable portions, is much more practical to implement than bringing God's kingdom down to earth. Quite possibly, the two programs are effectively the same.

Postscript:

The biblical Hebrew root תקן appears only in Kohelet, where it means “straighten,” “improve," or "repair." 

However, in Daniel 4:33, we find the Aramaic root תקן used in a different sense; that of "establishing a kingdom" -- as it happens, the very same sense used in Aleinu:

בֵּהּ־זִמְנָ֞א מַנְדְּעִ֣י ׀ יְת֣וּב עֲלַ֗י וְלִיקַ֨ר מַלְכוּתִ֜י הַדְרִ֤י וְזִיוִי֙ יְת֣וּב עֲלַ֔י וְלִ֕י הַדָּֽבְרַ֥י וְרַבְרְבָנַ֖י יְבַע֑וֹן וְעַל־מַלְכוּתִ֣י הׇתְקְנַ֔ת וּרְב֥וּ יַתִּירָ֖ה ה֥וּסְפַת לִֽי׃

There and then my reason was restored to me, and my majesty and splendor were restored to me for the glory of my kingdom. My companions and nobles sought me out, and I was reestablished over my kingdom, and added greatness was given me.  

We also see the Targums using the Aramaic root תקן consistently to translate biblical words with the root תכן or כון. See below from Alexander Kohut's Arukh HaShalem:

Simply put, though confusingly, תקן is the Aramaicized Hebrew (later, the rabbinic Hebrew) form of תכן. The Ashkenazic spelling לתקן in Aleinu is based on the Aramaic translation of the Hebrew תכן -- i.e., "establish” rather than “repair.”

I believe this also explains the word metukan in the Emet Ve-Yatziv prayer, which follows the Aramaicized spelling and should be understood as if it were the biblical Hebrew תכן:

וְיַצִּיב וְנָכוֹן וְקַיָּם וְיָשָׁר וְנֶאֱמָן וְאָהוּב וְחָבִיב וְנֶחְמָד וְנָעִים וְנוֹרָא וְאַדִּיר וּמְתֻקָּן וּמְקֻבָּל וְטוֹב וְיָפֶה

Mitchell First further pointed out to me that וְשמְּחֵנוּ בְּתִקּוּנו from the Mussaf Amida of the Three Festivals is the rabbinic Hebrew usage of תקן, also with the connotation of "establish."