This is entirely and old thing. Every private company runs on publicly funded infrastructure. Roads, sewers, etc...
There is no applicability of the First Amendment with respect to what a private company may censor. This is why Trump may talk about "freedom" but his executive order will not be grounded on any First Amendment claim.
Rather, Trump will essentially force the interpretation that CDA 230 should only apply to "politically neutral" actors. The result will be that social media companies will not be able to "censor" content that might be political. Again, not because of the Bill of Rights but because the liability shield provided by CDA 230.
The practical result will be that all content that is not clearly illegal (and potentially content that is) will be fair game and unable to be censored as any censorship will risk a claim against political neutrality and jeopardize CDA 230 protection.
The intent of the 1st Amendment applies most strongly to political and religious views. Liberals love it when the 1st Amendment is used to defend communism, terrorists, or profane art. If Communists and Antifa were being squashed by Twitter, they'd be playing victim.