Yes, but that also requires adopting a rather expansionist idea of corporate power and freedom of speech, even compared to post-Citizens-United America. Restricting a private actor from removing content they don't like creates a positive duty to publish content. For example, Net Neutrality regulation was legally opposed
specifically on 1st Amendment grounds, and that was a far looser regulation than what would be necessary to prohibit social media companies from making political decisions about what to publish.
More concretely, YouTube has actual knowledge that extremist right-wing political views damage their brand reputation - they've gone through several waves of pulled advertising that has harmed both themselves and their relationship with their creators. Advertisers have told them in no uncertain terms that they will not have their content on the same platform as, say, KKK rallies. So now we need to extend this positive duty to publish onto anyone who does business with these platforms, in order to prevent them from exercising the power of the purse to soft-censor views they don't like.
Furthermore, even this "duty to publish" standard for free speech does not always seem to meet the standards of some. I've heard people (not necessarily you, so the following is a strawman) argue that public rebuke is a form of censorship; if only because it may cause adverse publicity to have one's views opposed. This is absurd; the prescribed answer to offensive speech is counter-speech. If people decide not to do business with someone because they find their political views offensive, that shouldn't automatically be treated as censorship. Even at a "duty to publish" level (what you actually seem to be arguing for), you need to be careful to define common carriers narrowly to avoid forcing people into unwanted relationships, lest you run into this trap of "speech is consequence-free".