Free speech is a far larger concern than what the First Amendment covers.
Platforms that have grown into de facto public squares are large enough that there is a public interest in how they regulate speech, given that the vast majority of people are on the big tech platforms while the alternatives are negligible and choked out by network effects. They are now a gray area between state and what is typically understood by private entity.
There are massive and obvious dangers in letting Google, Facebook, Twitter decree for millions/billions of people what counts as "science" or "misinformation", and they've already demonstrated getting it spectacularly wrong. It's vital to a free society to have a culture of open debate, and these platforms are now our media for such debate.
It's not clear yet what the best mechanism or remedy for this will be, but citing the First Amendment alone is not sufficient to resolve the concern. The First Amendment only applies to government, and the cultural and social issues around free speech have always been much broader than that.
There are older precedents for regulating speech on private platforms in an earlier technological period: https://en.wikipedia.org/wiki/FCC_fairness_doctrine. Perhaps we need an updated version of that. I wouldn't be surprised if the platforms themselves were ok with it, since I don't believe they wanted to get into the truth-regulation business in the first place. They're doing it because they're subject to intense, inner and outer, political pressure.
Without getting into rightness or wrongness of views, I think it's factually the case that those pressures are coming from a fairly narrow band of political opinion which is far from being the entire spectrum and excludes many mainstream views, so in that sense what we're seeing is something analogous to "regulatory capture" of the big platforms. This gives enormous leverage to one cluster of political opinions over all the others. It's a distorting factor which I think is against the public interest (even though I personally agree with some of their views).
We need debate and persuasion, not hard use of power. This understanding was commonplace, in fact bedrock, for so long that many of us are shocked at how quickly it could be abandoned, and not just abandoned or even forgotten but replaced with a narrative in which it was never that way to begin with!