The difference between a lie and a mere falsehood or even a mistake is intent and knowledge. The legal system is perfectly capable of determining intent and knowledge (when including not just actual but cases where knowledge was recklessly disregarded). We also already have laws that in certain contexts outlaw such lying. The most common examples are perjury and lying to officials either FBI or immigration, the latter specifically in the context of one's status. The laws, particularly the first two, lost a lot of their punch because of rampant misuse by authorities - the latter overused by federal agents to buy time to build cases they can't otherwise Constitutionally establish legally, the former was found to be so common among police testimonies that it had both a term - 'testilying' and hundreds of academic papers on the subject leading to no reform and little added enforcement. We also have laws against defrauding - which usually just adds onto a specific intent to gain something of value. It's not that we can't keep free speech as the Constitution sets it but still regulate particular instances where the state can prove a case of bad intent and knowledge, but the long history of applying these laws not for justice but for coercion, to fill quotas, carelessly, and unevenly have eroded the erstwhile authority prosecutors have in using them without creating a political issue for him or herself. I think restoring integrity in the system so that these cases can be taken as nonpolitical is arguably an even more difficult task, but certainly if done right, would have the most legitimacy and be the most narrowly targeted.

But outside of that I'd much rather have corporations than bureaucrats handling such matters, because at least they can be incentivized to do the right thing. After all, corporations are ultimately voluntary entities formed and owned by average people, and when their conduct fails to match societal expectations, they lose customers and revenue, which is at least some incentive. Furthermore, they're protected by the constitution to do the moderating. They are allowed to keep the undesirable off their platform. No matter how much Josh Hawley wants to nationalize big tech to make America into White China, he can't actually do it without an armed mob. In contrast our administrative laws and particularly those with enforcement powers have effectively been turned into the Constitutional backdoor that through sheer luck Trump was unable to realize as the easy and if not legal, at least unchallengeable way to stage a coup, just as their suppression and kidnappings over the summer resulted in no real consequences, it's by design, and one that circumvents much of the Bill of Rights with essentially no oversight, judicial or congressional, the former by doctrinal custom and the latter by negligence. They pose a much greater danger, because they not only get into risky situations, legally they're effectively immune from almost all responsibility in a way that even puts qualified immunity to shame. I hate to pose these not as hypotheticals but as actual comparisons, but Apple haven't kidnapped me for all the complaints I made about the shitty 2017 MacBook keyboard yet, while federal agencies with militarized regulatory enforcement powers have, supposedly all through mistakes, "deported" over a thousand American citizens to random countries they have no connection to. They're the backstop we should never lean on.