Jack, yes, Justice Windowdressing's first name is indeed "Secret"; no member of the public can tell how he really feels.
Driving my hypothetical is of course a question about what legitimacy means, and I'm not sure you have answered it. The problem is that legitimacy to the public and legitimacy among the Supreme Court cognoscenti are two different things. I'm not entirely sure which you have in mind.
Take me in 1993, for example. Fifteen years ago, I was an engineering major in college and had only a passing interest in the law. I read Linda Greenhouse's descriptions about the Supreme Court in the New York Times because I read the Times every day, but I had never actually read a Supreme Court opinion.
In 1993, did I think the Supreme Court's decisions and caselaw were legitimate? Of course! To be candid, it never dawned on me that they couldn't be legitimate. The justices were the nine most brilliant legal scholars in the country—that was why they were on the Supreme Court, obviously!— so it seemed absurd to me that anyone could even think to question that they were right. I might or might not like their decisions, but they were the oracles of the law, and I was just an engineering student and obviously they knew best.
My sense is that when you use legitimacy, you don't mean legitimacy to someone like the Me of 1993. That would set the bar pretty close to zero; just wear robes and hear arguments in that fancy building, and you'll trick young Kerr. Rather, you seem to mean legitimacy to a group of lawyers somewhere, or maybe a law school faculty, or maybe some combination of these various audiences. That's why I asked you whether any real-world justices have not generally followed your advice; I was hoping your answer would help articulate the relevant audience that assesses legitimacy.