The idea that it is somehow the religious duty of Jews to "repair mankind" is not only unfounded it is a manifestation of the ignorance of assimilationist Jewish liberals. The simple fact of the matter is that in actual Judaism, it is none of the business of Jews to fix or repair humanity. More generally, in Judaism it is the job of Jews to repair the Jews - a not inconsiderable task - not to repair the world.
Non-Jews are not in need of being "repaired" by Jews, at least as long as they observe the seven "Noahide Commandments," the rules of living that Jews interpret to be conferred upon all humans, all descendents of Noah, by God. Beyond that, what the gentiles do and how they do it is none of the business of Jews, and Jews simply have no religious standing to interfere.
It is certainly not the job of Jews to instruct non-Jews about matters such as income and wealth distribution, abortion, environmentalism, health care provision, or discrimination. Only in matters of cruelty that negate the Noahide Laws are Jews commanded to interfere.
Indeed, the very notion that Jews are so ethically superior that they are entitled to instruct non-Jews in ethics is completely foreign to Judaism. The self-image of Jews in the Torah is that of a group of people awash in their own moral failures and foibles, from the Golden Calf to the paganism of the era of the kings of Judah and Israel. The moral imperative of the Torah is for the Jews to improve and reevaluate their own behavior, not to pretend to have the moral superiority to preach to the entire non-Jewish world.
"Man" may very well be in need of redemption and improvement and repair, but it is not the business or job of the Jews to carry these things out. And it would be hubris to think that Jews are morally equipped to do so. Jews have more than their hands full in attempting to repair Jews.