I don't mean to cast doubt on the tweet's purpose (I agree with it), and for what it's worth I've emailed other researchers for paper access as well. I would just like to see some proper legal analysis behind these questions, because I see the same point repeated frequently by scientists who are not lawyers.
It would be pretty cool if there were a platform that automated your request to get a paper from an author. I don't know how that would be plausible without hosting the papers though, and I imagine that would get into trouble.