I don't follow the logic that criticizing Trump's stupid tweets is an attack on free speech, or follow how "my logic" makes that argument.
But, I do believe that braindead hot takes about the New York Times ("NYT is trash," "NYT should retire," and other typical brainless mouth breathing broad condemnation of prominent newspapers whenever there's any sort of ... just dumb article or take by one of its writers) that are posted thoughtlessly and generally go unchallenged are part of a larger motive to break down the perceived integrity of the best American journalistic outlets, yes. The conservative right has been doing this for decades, which is why it's so easy for Donald Trump to go on Twitter and declare that the NYT is the enemy of the people, or that the Washington Post can't be trusted because it's owned by Jeff Bezos, or that any publication that writes critically of Trump is "fake news" or lying, or what have you. The Right has had an assault on truth for decades and it's given rise to a wannabe authoritarian who has taken over about 40% of American political ideology. I'm always happy to counter-balance brainless attacks on the left of the New York Times, because I don't want the same thing to happen to my political ideology in 5, 10, or 20 years.
In this community and some of the other communities I'm a part of (Reddit, Twitter, YouTube to an extent) saying "The New York Times is a good newspaper" is usually a more controversial take than "The NYT is Trash." I'm okay being that counter-weight.