I think it could require government action: I could argue that it was government inaction in allowing a monopoly/oligopoly over the conduits of free speech that is now depriving me of my rights.
For what reason are we considering these platforms "conduits of free speech"? They are simply private services, private property. In the same way you can legally remove people from your home that you don't want in there, they can bar you from their service. They also didn't always exist. At what point in their existence would you argue that being banned from being able to use them was depriving you of some right? Can they have a monopoly over all of the "conduits of free speech" if all the old methods of communication still exist? If those alternatives exist, could they really be called a monopoly? (Although for things like Twitter, they're definitely not a monopoly, but if we're talking about govt backed ISPs, which can be a monopoly, then that is indeed a different story, but I would argue that ISPs should be divorced from any government regulation/subsidies).
"In the same way you can legally remove people from your home that you don't want in there" -> I agree with you in principle, but this is the type of thing where the principle doesn't generalize at every level of scale, and at a big enough scale it becomes problematic.
Let's consider the other extreme with a fictional corporation "MEGA INC", which suppose owns all web hosting, all ISPs. Let's also throw in that they have a monopoly over paper production and publishing. Now, do you think your argument that "this private entity can do whatever it wants" is problematic? I should hope so.
I'm not making the case that it's black/white and that this situation with Big Tech is equivalent to MEGA INC. But, it's not that our free speech rights are binary. My point is simply that we're somewhere along the spectrum spanning "private home" <-> "MEGA INC", and at this point rights are actually being diminished because of the oligopolistic nature of a significant corner where discourse happens.
So, unfortunately, I think it's a nuanced situation that's not easily reduced to a simplistic principle such as what you've stated. We have clear principles to reason about the extremes, but it's hard to make an argument in the hairy middle because both can be made to apply.
There's no shortage of conduits for free speech you have a right to use, on and off the internet.
The oligopoly only concerns distribution to the widest possible audience, whether it's social media or broadcast media. And even a First Amendment constrained government is allowed to pick and choose which speech it distributes