The ICC is not a beneficiary of the Constitution, nor is YouTube bound by the Constitution. I'm unhappy for the same reasons as you, but this isn't how 1A works.
nitpick - Youtube is bound by the US Constitution, it is the highest law of the land. 1A[1] is only about binding the government/congresses power though so youtube is not bound by 1A.
The US government has effectively ordered YouTube to take down these pro-Palestinian YouTube channels.
When the government pressures companies to censor Constitutionally protected speech, that is a First Amendment violation. If it weren't, the First Amendment would have no practical meaning.
The problem is that these private companies have taken a disproportionate place in public discourse. You are absolutely right that freedom of speech does not guarantee the right to post anything on YouTube (someone else's website). In fact YouTube has the right (protected speech) to censor you and refuse to let you post long as they don't do in a discriminatory way (for instance, only "white people" can post would be discriminatory/illegal).
The problem is that in practice, if you can't do YouTube, Facebook, Tiktok, INsta, etc... your speech will not be heard by anyone. It's like if a tree falls in the forest and nobody is there to hear it, the fact that it makes sound is irrelevant. So effectively, it amounts to censorship, even though the government potentially had no hand in it.
Now imagine someone in Trump administration pressured Google with a juicy contract, or the prospect of an expensive lawsuit, and the quid pro quo was dumping these videos that annoy "our Israeli friends". This kind of "pay to play" is at minimum corruption. It may also fall of short of constitutional guarantees for free speech. Ironically, it is exactly the same thing a lot of members of the Trump administration have accused Biden of doing (exhibit: the so called "Twitter Files" etc... ), although I don't believe this went anywhere in federal courts (am I wrong?)
I honestly don't know what the answer is. But I would not be surprised if in 50 years time, some of these large companies get regulated as "utilities" and are no longer able to yank "videos" from their platform just because they feel like it. And every time they "abuse" their powers, I feel like we get an inch closer to that onerous regulation.
