Free speech is an American value, not just a few words in the Constitution. I'm sympathetic when folks object to private censors. As an American, I'm offended as well.
The “platform” excuse is dead on the first real legal challenge. It’s past time for these publishers to be held to the standards other publishers have been held to for decades or longer. They can’t cry “free speech” on one hand while acting like publishers on the other. When it was just galling hypocrisy fair enough, but they’re big enough and have supplanted enough “traditional media” that their time has come.
The post office is more analogous to the transport layer, in which case you'd be making a solid case for net neutrality, and I'd agree with you there. The post office having an editorial board would be like Comcast having an editorial board.
The problem isn't with excluding "X" screeds. The problem is with dishonest actors who happen to have corporate power who label everything they disagree with as "X." The problem is with arbitrary corporate media power which can have an effect commensurate with government violation of the 1st amendment, but which skirts the law because it's "private."
In the 1960's the government sought to suppress the speech of students and large segments of the populace, then that populace sought to fight back using Free Speech. Now, in the 21st century, corporate media and tech power seeks to suppress the speech of large segments of the populace online. The question isn't whether or not it's the government. It's the question of whether or not coercive power is going against the will of a segment of the populace. It's the question of force vs. argument.
The issue is not the government shutting down media. It's with big tech shutting down small mom & pop media. If it's wrong for the government to censor, we shouldn't let the authoritarians skirt the constitution by gathering the commensurate power (in the form of a monopoly) to do just as much.
"Or tell them what to say." Is there anyone who actually thinks that what some random YouTuber says is the official word of YouTube? They probably exist and comprise only a sliver of a fraction of a percent of the population. Censorship on a media platform should not be a form of speech. The fact that this is such a sticky issue probably indicates that another category is needed.
If the government tells YouTube which YouTube videos to suppress, no matter how "unimportant" the "random YouTuber", then the government is restricting both the YouTubers' speech, and YouTube's right to prioritize information as it sees fit. Just the same as if the government told newspapers which articles were allowed to appear on the front page, or which op-eds it could select.
Let's turn your criticism right back around: Would that kind of regulation be tolerable to you if we were talking about conservative publications/websites, and it were a liberal government telling them what to do?