This isn't the argument I made, but I will make one point along these lines. It's probably more effective in terms of activism. However, what about the dissemination of ideas? It used to be the most important thing for the dissemination of ideas was the printing press. Of course there was a time before the printing press, but that is besides the point. The printing press allowed for ideas to have far greater reach. So important was the press in those times that people argued for 'freedom of the press'.
Now we have the internet which gives those ideas even greater reach. It is only natural then that people would argue for 'freedom of the internet' right? Recognising it's importance. That's not to say we're in real trouble now, necessarily, but I will at least say the trend is troubling.
Freedom of the press, then and now, does not mean anyone can say whatever they want and call it press and then it's suddenly this sacred fourth estate. Alex Jones may have the protections afforded the government, but he is not "press".
Also, Freedom of the press is a still just a guarantee from the government. Our institutional organizations decided that the protecting press and speech from interference is an important aspect of democracy, but they didn't explicitly say "people can say and print whatever the fuck they want". We have laws that restrict speech and press.
Freedom of the internet is already achieved under Freedom of speech. The government cannot arbitrarily decided what is and isn't on the internet, though like our 1st amendment, there are limitations. Freedom of the press does not necessitate that a 7-Eleven is constitutionally required to sell USA Today, just like Freedom of speech does not necessitate that YouTube is constitutionally required to host Alex Jones.
It may not have meant that in actuality, but that is exactly what people argued for. The press has been around for a long time. It's like saying freedom of the press does not mean you can say anything heretical against the church. Not in actuality, but it meant exactly that.
And in terms of freedom of speech more broadly, I think there is a small problem with the 1st amendment. Don't get me wrong, I am jealous you have such a thing. It is an incredibly useful document. What it does mean though is people in the US often argue in terms of the 1st amendment and not necessarily in terms of freedom of speech as a principle. For instance a freedom of speech absolutist would argue against any censorship, government or otherwise. Not strictly because of the 1st amendment but as a social principle. It would be inconsistent to say for instance that you're a free speech absolutist, but yet you personally are against the principle of free speech when it comes to the institute you belong to. No they're for free speech within that insitution as a matter of principle.
One of the greatest free speech advocates, John Stuart Mill, didn't just argue free speech solely when it came to the state, but also noted other forms of censorship. In fact most of the great arguments for free speech does not address it solely as interference by the government but notes other forms.
However, as I have already ackowledged, freedom of speech does interfere with property rights. So ultimately is best not to advocate for compelling it, but rather encouraging it. To try to foster an environment where freedom of speech can thrive. It is no use for instance if you have free speech protections from the government but all private enterprises operate in a way that it is
as if there is no free speech protections from the government. It may not be entirely the same thing, government censorship is far war worse than corporate censorship, but if the result is the same either way, you can at least see there is a 'problematic' aspect to it. That's not say things are that bad in reality, it is just a hypothetical that demonstrates it can be still just as problematic regardless of whether it is done government sponsored or otherwise.
I will also point out that the threat doesn't come solely from the US, Europe is just as much a threat, possibly a bigger threat since there are no free speech protections.
