As the link you provided states, this decision applies to shopping centers in California, whose state supreme court has narrowed its applicability a few times over the years.
Read the appeals cases. They curtailed spaces like Costco parking lots and strip malls without plazas or atriums, which are different than "common areas". Free speech in common areas was reaffirmed in 2012.
YouTube isn't even an area, much less a common area. (Not to mention that the “common area” thing is not a federal Constitutional requirement but a judicial application of the positive rights in the California Constitution; it is not a First Amendment right.)
It's a publication in which user submissions that Google accepts will be published, possibly accompanied by ads from which revenue is shared with the submitter.
This is a weak argument, nitpicking semantics. Virtual spaces are protected, too. [1] Telephone systems and television cable systems are not technically "areas" either, yet have protections under the first amendment.
Furthermore, you contradict YouTube's own mission statement:
> Our mission is to give everyone a voice and show them the world.
> We believe people should be able to speak freely, share opinions, foster open dialogue, and that creative freedom leads to new voices, formats and possibilities.
> We believe everyone should have a chance to be discovered, build a business and succeed on their own terms, and that people—not gatekeepers—decide what’s popular.
> Telephone systems and television cable systems are not technically "areas" either, yet have protections under the first amendment.
Users of telephone systems have free speech protections against the operator of those systems not because of the first amendment (and especially not because of applications of that to physical common areas against private property owners, which applications don't actually exist—the First Amendment has specifically and repeatedly been held not to apply even in that case against the property owner), but because of common carrier regulations.
> Furthermore, you contradict YouTube's own mission statement:
YouTube's PR has very little impact on how constitutional law applies to it.