LB, I’m not sure and there are probably people here who know better, but I will tell you what my impression always was:
I thought that it really is a Torah concept (in the paragraph after Aleinu, we say “l’sakein (l’takein) olam b’malchus Shakai…” which means “to fix the world with the Kingdom of Hashem”), but the secular world adopted the concept and gave it a broader, more secular, liberal meaning (and basically changed the meaning) so the Frum world doesn’t usually use the concept since it would be misunderstood according to the Secular Liberal translation.
Note that we talk about: “l’sakein olam b’malchus Shakai”, the world can ONLY be fixed (receive a tikun) through Hashem’s Kingdom, and I think that part is probably left out of the secular interpretation, whereas we would say that it’s the main part.
Regarding Avi Weiss, since he is no longer considered to be Orthodox, his usage of the term is really irrelevant.