Libertarians don't believe the legal system should be set up so that rich people can use it to censor speech they dislike—even if that speech is utterly reprehensible, as it is in the case of Gawker.This is obviously untrue: of course libertarians support a legal regime that the rich can use to censor everyone else. For example, libertarians think that it's okay for the rich to coerce workers into contractually waiving all kinds of speech rights. If the worker refuses to enter into this arrangement, libertarians believe that the rich should be able to use the government to enforce all kinds of property laws against the worker until he starves to death. If the worker tries to break the contract, libertarians believe the rich should be able to use the government to prevent him from doing so, or to violently discipline him.
Note that this isn't some kind of unlikely, edge-case scenario - it's just a technical description of a situation that most of us are already in. Most employers have all kinds of expectations and even formal policies about what employees should and should not say. And most of us work for such employers, not because we want to waive our speech rights, but because we need to eat and pay the rent.
The libertarian obviously rejects this description. For instance, he insists that true capitalism would not be an economically coercive contractual regime; no one would actually face take-the-job-or-starve-dilemmas, because we can always just become entrepreneurs or live off the land. Perhaps this is true - but suffice to say that the censorship argument is now just a proxy argument for the real underlying disagreement, which is about economics. The leftist believes that capitalism inevitably creates a power imbalance between the bourgeoisie and the proletariat, which gives the rich the opportunity to censor the poor; the libertarian believes that capitalism can facilitate a "free" labor market where economic coercion cannot exist, which means that no one can be coerced into waiving their speech rights.
This puts Soave in the exact same boat as Menaker on CTH. Both insist that the government should enforce a particular legal-economic regime that would maximize the freedom of speech. Both insist that, short of that ideal, we should selectively approve of interventions by the government for the sake of justice. Menaker prefers interventions through our judicial system against the American right; Soave prefers interventions through our property and contracts laws against exploited workers. If this assessment of their disagreement is wrong, it's wrong because of fundamental premises about economics - but it's obviously not wrong because of differing commitments to free speech.