“Mainland Asia” makes a certain amount of sense.
“Mainland China” might make sense as long as we didn’t think it included Hainan.
“Chinese Beijing” is not quite right. To make the analogy perfect, we’d have to call it “Japanese Beijing.” (Actually, the PRC Olympic team was supposed to be called “Chinese Beijing”–short for “Chinese Olympic Committee, Beijing”–but those Communist Canadians liked them better than us.)
Taiwan is “the mainland” for people living in certain of its outlying islands.
Don’t Hawaiians call the continental USA the “mainland”? That’s a lot farther away than Australia and New Zealand.
You know what would be a good analogy? India and Lanka. (I refuse to call it “Sri”.)
Funny that people stopped saying “Free China” about the time it actually became free.
Rather than an “undercurrent,” I would call it a small group of activists. They have a good case from the perspective of international law, but not so much from the perspective of political reality. Also, I don’t see how they get from “Taiwan really should have been given a plebiscite” to “We are the true government of Taiwan.” It’s a bit like declaring yourself king.