Well given FTL sensors, i think people might notice if the Mutura Nebula suddenly disappeared.
We don't know the range of the sensing systems in the TREK universe and how commonplace they may be. Professional astronomers and researchers might have access to starship-grade tools, but would ground-based astronomy, as we know it, still exist in a universe with fleets of starships? (Why fund it when a starship or unmanned FTL probe could get better information?)
Given time, what happened in Mutara might become known, but how many light-weeks, -months, or -years away is the nearest inhabited world that would see it? It's more likely that people would find out from leaks or hacks of the Starfleet databases.
The only thing the alien in the bar knew was that "Genesis planet is forbidden." Talos was a forbidden system. Maybe Starfleet had found something just as dangerous and warned all travelers to avoid the area.