Journalistic integrity is therefore critical when selecting which biased sources to pay attention to.
1. The arguer claims that negative consequences follow from the exercising of free speech, in this case NYT right to freely chose the topics they write about.
2. The alleged consequence is that people are made to believe wrong or false things (where "wrong" and "false" are defined by the arguer).
3. The arguer portrays himself at the same the victim of those media and the person who knows better than those media and therefore can decide between wrong and right, true and false better than the accused media.
4. The arguer presents no evidence of knowing better and when you ask them about their sources, they tend to be highly problematic, based on blogging and websites who often do not even employ journalists.
Paraphrase: "I know better than large group of people X but everybody else is mislead by X" - I don't think so.
If you write a blog post that doxxes a prominent figure and link to it from Facebook and Twitter, you are going to get banned from those platforms. The NYT can apparently do this with impunity, and calls for canceling other people and organizations who do this.
In US law, there is a different standard for libel against "public figures" than against other people. The NYT gets to take advantage of this much looser libel law whenever they write a hit piece because they can argue that anyone who does something "newsworthy" is de-facto a public figure.
As far as I have seen, the "cancel NYT" crowd is arguing that the NYT should be held to the standards that it pushes into others and obviously doesn't follow.