Being a partisan mouthpiece isn't itself a problem, the issue is when it pretends (and many of its supporters repeatedly and falsely claim) that items described in the paper are more objective and carries greater weight than those in your average political party's weekly newsletter.
Everybody is biased. You, me, and every journalist on earth. Of course, that's okay. The NYT also does not go out of their way to "...pretend they have no political bias."
What is important is to be able to understand the difference between news and editorials (including editorial decisions), but sadly more and more people seem to lose grasp of this basic distinction. This may be a sign of the negative consequences of the politization of many points.
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.
Political lean != acting as a mouthpiece.
Do also please note that your personal political leanings will determine whether you view the reporting of any publication as unethically biased or not. No matter which sides we're talking about, what one party reports as truth, another will hear as politically motivated.
With social media and modern communication/publication mechanisms, it is much easier for individuals who know the ground truth to bring their perspective to the fore and poke holes into a major publication's journalistic flaws. This wasn't possible just 10 years ago.
In the case of NYT, their political tilt is very clear (that's ok) but their journalistic integrity is being called into question more and more (that's problematic).