Where there's less room for debate, responsibility includes an obligation to avoid "shape of the earth: views differ" approaches.
If individuals and institutions don't take up this kind of responsibility, then there's no such thing as discourse, only speech as indulgence.
And the value of popular opinion is directly related which one of those is used in constructing it. Individual and institutional responsibility, however, will remain orthogonal to that.
For anyone else to insert themselves into the conversation, public or not, and claim to have The Real Facts is arrogant. For an institution to attempt to referee the conversations it is privy to, prevents actual truth from being discovered. To require institutions to referee in that way? Is another act of arrogance: placing oneself above 'the other people who have to be kept in line'. To leave those institutions at the mercy of popular opinion, to referee as the mob sees fit, is insanity.
At least, that's the way I see it. Thank goodness we're still allowed to disagree on this.
Facts are often debatable; the set of facts that need to be accurate is also debatable; and you are suggesting that both of those debates be settled unilaterally by expecting (requiring?) the institution to "take responsibility for accuracy." Note that public pressure (read: popular opinion of the day) will almost certainly be the deciding factor in how those debates are settled by those institutions.
Are you suggesting that it is wrong for institutions to abstain from "bowing to the popular opinion of the day" in how they censor speech?