No, the "public square" designation specifically allows people to do things like come to the mall and set up political protests and whatnot. It's not about _access_ to the mall; it's about speech protections.
Is uploading videos more akin to setting up a store at the mall, or more akin to standing on a soapbox at the mall and giving a political speech? It probably really depends on the video, on whether the video is being monetized, etc.
In particular, I feel there is an important distinction between demonetization and removal here, from an ethical/moral perspective. I can't speak to the legal perspective; I am not a lawyer.
If a mall did that they likely be accused of discrimination. The demonstrators don't demand to be promoted, but simply be treated as anyone else.
That said, I'm not quite sure I understand your point. A speaker in physical space can pretty easily take their speech elsewhere; most obviously to the nearest public street corner. So moving away from a private venue typically does not require anything nearly as drastic as "moving to another state/region".
I'd really like to understand your point and how it applies to both the mall and youtube situations, though, and would appreciate you explaining it if you have the time.
I think there's a difference between "you can speak" and "you can speak in such a way that interested people can hear". The former is not very useful in terms of the right people usually think of as "freedom of speech"; the limiting case of it in the physical world is "you can speak, but only in your own home". So what, if anything, makes for an online version of the public square, where one can go to present speech for consideration by others?
Also, I think online speech is more similar to physical-world speech than you make it out to be. You can't speak online without "imposing" on your hosting provider, their ISP, etc. If you self-host, you "impose" on your own ISP (and probably violate their ToS, if you have a residential connection). You "impose" on your domain registrar. These are all private entities, so you have the same sorts of issues as you allude to for the physical world. And these private entities have been known to restrict the speech of people relying on their services, so this is not a hypothetical risk.
The extent to which someone is entitled to an audience is of course more difficult to judge.
That's the idea those absurd behind "free speech zones" off in a remote corner.
Free speech means free speech in every public space.
Personally I fear the future where the government (or monopoly/oligopoly) in power isolates all opposition in a remote corner where no one can hear.
Even then, it does not mean that those malls should allow anyone to sell wares. What it means it allows people to visit without discrimination on basis of legally protected classes like gender and race. YouTube banning a channel does not ban individual from watching videos, it prevents them from uploading videos which is akin to setting up stores.