Personally, I would go with wikipedia, craigslist, twitter, internet archive, and a public search algorithm. What else should be added?
Or put another way, why twitter and not a "public short message platform"?
Your comment is likely targeting public "utilities" on the web. Of the ones you mention, I only think search and archive are "the utilities of the web": wikipedia gets close, but it's a summarised version of the archive.
Basically it should be a product that is over seen by a group of diverse individuals with different needs. All these mega sites have the goal of making as much profit, as fast as possible which is causing a lot of chaos in society.
Well none of these should be public domain, since were created, and are run by, existing organizations. Going down the route of "if you become successful, you'll get "nationalized" and placed in the public domain" seems to be, well, not likely to get much support.
Of course nothing stops you creating a project, placing it in the public domain, and then driving the project, getting funding, attracting an audience, becoming a meaningful social good. (See SQLite as an example of this approach.)
But turning over existing, valuable, successful projects to a "Trusted Public Body" (whatever that is? trusted by whom? which public? .org debacle anyone?) is simply a good way to kill successful projects.
And that's before we get to impossible questions like "who decides what gets moved to public domain?" - the Greek government? Silicon Value investors? my mom?
Information monopolies should be federated.
Academic research should be public