The claim that he owns the account is not the same as claiming to own its contents. E.g. a landlord that owns an apartment typically does not have a claim on their tenant's property in the apartment, except under special circumstances. They may have some liability for their tenant's property in some cases, e.g. if they knowingly allow a bomb factory. But Section 230 is an explicit waiver for that liability in terms of speech.
Both parties are very interested in limiting the scope of that waiver in different ways, but that's the current law, and it tends to be broadly interpreted.
This would then also imply that any post containing illegal stuff is also his problem, wouldn't it?