See, YT bans all kinds of stuff that the first amendment protects, including pornography, violence, and a lot of other nasty stuff that they don’t want on their platform for entirely clear reasons. Eliminate YT’s right to clear those off their site, and it’ll be filled with the lowest common denominator very quickly, as regular users flee from the torrent of unwanted offensive material.
Of course, it’s easy to imagine that we can give YT the right to ban some “offensive” material, but who defines what “offensive” is? The Government? Giving the government the right to control what can and cannot be moderated on YT is kind of contrary to the entire point of this exercise.
If you believe that YT controls too much of the information in our society, then anti-trust or anti-monopoly law is a much richer area for fostering free speech.
Aside from that, removing the first amendment from my argument changes very little. You’d still be obliged to create a definition of what is and is not free speech and a mechanism for enforcement, which would be subject to the pitfalls listed above.