It's a Chabad thing. The Rebbe held there is a Chiyuv to be mekarev goyim to keep 7 mitzvos. I've been curious for a while what the source is, since as you mentioned there is no הוכח תוכיח and no כל ישראל ערבים זה לזה.
@PlatinumGuy https://asknoah.org/wp-content/uploads/lubavitcher-rebbe-noahide-campaign.pdfThe Sheva-Mitzvot Campaign
Torah obligation: On what basis? What is required?
a) Moses was commanded by G-d at Mt. Sinai to compel all inhabitants of the world to accept their Seven Mitzvot.- Rambam Laws of Kings 8:10; Hisva’aduyos: 5746 v.2, p.43-44.
b) Every Jew has a mission to influence the Gentiles to behave in a manner consistent with Isaiah 45:18: G-d formed the world to be settled. This is through observing their 7 Mitzvot and all the offshoots that follow from them… - Hisva’aduyos 5744 v. 3, p. 1538; 5745 v. 3, p. 1450-1451.
c) A Jew travels to a place “to establish His Name there” by bringing Gentiles to express G-d’s glory: accepting His kingship willingly, and learning about the Noahide Code. - Hisva’aduyos: 5748 v. 1, p. 298-299; 5749 v. 1, p. 60.