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.
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.
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