This is a very good question and actually touches upon one of the fundamentals of Jewish belief.
The concept of 'Tzibbur' (ציבור), as in communal obligation does not relate to non-Jews. Tzibbur within the Torah is a unique characteristic to the Jewish people. That in addition to their status as individuals, they have another state of being where the collective of individuals is a distinct, unified, single entity.
This is expressed through the posuk from 2 Shmuel 7:23 which says:
וּמִ֤י כְעַמְּךָ֙ כְּיִשְׂרָאֵ֔ל גּ֥וֹי אֶחָ֖ד בָּאָ֑רֶץ אֲשֶׁ֣ר הָלְכֽוּ־אֱ֠לֹהִים לִפְדּֽוֹת־ל֨וֹ לְעָ֜ם וְלָשׂ֧וּם ל֣וֹ שֵׁ֗ם וְלַעֲשׂ֨וֹת לָכֶ֜ם הַגְּדוּלָּ֤ה וְנֹֽרָאוֹת֙ לְאַרְצֶ֔ךָ מִפְּנֵ֣י עַמְּךָ֗ אֲשֶׁ֨ר פָּדִ֤יתָ לְּךָ֙ מִמִּצְרַ֔יִם גּוֹיִ֖ם וֵאלֹהָֽיו׃
And who is like Your people Israel, a unique nation on earth, whom God went and redeemed as His people, winning renown for Himself and doing great and marvelous deeds for them [and] for Your land—[driving out] nations and their gods before Your people, whom You redeemed for Yourself from Egypt.
And this same idea is also explained in Brachot 6a in the name of Rav Nachman Bar Yitzchok who says:
Rav Naḥman bar Yitzḥak said to Rav Ḥiyya bar Avin: What is written in the phylacteries of the Master of the world? Rav Ḥiyya bar Avin replied: It is written: “Who is like Your people, Israel, one nation in the land?” (I Chronicles 17:21). God’s phylacteries serve to connect Him, in a sense, to the world, the essence of which is Israel. Rav Naḥman bar Yitzḥak continues: Is the Holy One, Blessed be He, glorified through the glory of Israel? Rav Ḥiyya bar Avin answered: Yes, as indicated by the juxtaposition of two verses; as it is stated: “You have affirmed, this day, that the Lord is your God, and that you will walk in His ways and keep His laws and commandments, and listen to His voice.” And the subsequent verse states:
“And the Lord has affirmed, this day, that you are His treasure, as He
spoke to you, to keep His commandments”
(Deuteronomy 26:17–18). From these two verses it is derived that
the Holy One, Blessed be He, said to Israel: You have made Me a single entity [ḥativa] in the world, as you singled Me out as separate
and unique. And because of this, I will make you a single entity in
the world, and you will be a treasured nation, chosen by God.
This idea is also discussed by the Rogatchover Gaon in connection with the reasons behind why the judgements against the Rebellious City are different from those of the individual, even though the transgression is the same. For Jews, the Community, meaning Tzibbur, is a distinct entity.
The Lubavitcher Rebbe also discusses this concept, quoting the Rosh, in connection with the subject of ציבור לא מת, that the Tzibbut does not die. See also Yachin to Mishnah Temurah 2:16:1, Tosafot Yom Tov to Mishnah Temurah 2:2:3, Steinzaltz to Temurah 15b:6, and elsewhere.
The mitzvah for the Noahide is not to fail to establish courts of justice. The function of these courts is to arbitrate disputes between non-Jews and to educate them about their individual responsibilities as members in a civilized society. All of this promotes the general peace.
When you ask to what extent should the individual exert themselves in that direction, it would seem to be associated with the desired result. If an increase in unresolved disputes and the breakdown of society is being observed, it is an indication that there is a lack in establishing those courts of justice whether to resolve disputes or to educate the public about their commonwealth. It is clearly in their best interest as individuals when the society functions peacefully.
Regarding your discussion of whether modern day, secular courts fulfill the concept of the Noahide court, this would seem to revolve around who is a Noahide and who is not. Simply being a non-Jew does not qualify someone as being a Noahide. A Noahide acquires that status through meeting the legal definitions outlined within the Torah. In that context, none of the current, secular, civil courts among the non-Jews, that I am aware of, would meet that criteria simply by definition.