I think that's not as simple as you describe.
The glib answer is, "They publish corrections." Which many do.
But I don't think that addresses the (valid) criticism about the Iraq invasion: while there were some errors of fact, there were primarily errors of omission, where the media were too willing to act as uncritical stenographers for politicians' claims.
We see, in a way, the same phenomenon in the Trump era: whatever lunatic allegation Trump or his acolytes would make, the media felt the need to cover it; after all, whatever the President says is inherently news, no matter how crazy.
Yet in both of these examples, the problem isn't that the media represent some sort of powerful cabal, which I think was what the OP implied: it's that the media are too spineless to truly hold to account those in power (in both cases, the President of the US). Viewed that way, the premise of the original question is incongruous: it's deeply strange to criticize only the "Princeton and Columbia"-educated "media elites" who too-uncritically reported on the claims of the President, and not the Yale- and Harvard-educated President (and son of a President)--or the billionaire heir to a fortune President who came later--who uttered those false claims.
The media absolutely deserve to be criticized for their uncritical reporting before the invasion of Iraq, as I said before, but the original criticism sounded like we want a weaker media who will be less able to question false narratives espoused by powerful players. And I think if you look at these stories more closely, you'll realize we want the opposite.