No. Google is different. They hold a monopoly on search that I think should be broken up. I have big problems with the power Google wields in finding things online.
But, going back to my previous comment, it would in a way be a good thing if Google started capriciously "disappearing" sites like this, in a high-profile manner. It would accelerate the move to other search providers and bring a more diversified Internet to us all. (I'm not advocating for it, but the end result would probably be better than what we have now: SEO madness all to please The One Algorithm.)
Twitter has competition. Reddit, Facebook, Apple News, and traditional media outlets all disseminate information. They all offer notifications on your phone. Where we're typing this out is HN, which is another news dissemination and discussion service.
The more Twitter does things to make it clear it's not a neutral platform, the more other services can find users. I do not want Twitter to be a monopoly platform. Tagging them with some common-carrier-like status, while allowing them to remain private, just entrenches that monopoly.