Moreover, I've seen no evidence whatsoever that Elon Musk is looking to be considered a philosopher. He writes short meme tweets, not treatises.
Well, I agree that the concept of philosopher-king is applied by people nonliterally to Thiel, though mainly because those people are ignorant of Plato. But if we interpret your comment nonliterally and remove the "from Plato" part, I still don't see how the nonliteral concept helps to explain anything. Unless we're supposed to interpret "It explains a hell of a lot" nonliterally too?
As I see it, calling Thiel a philosopher-king wannabe is nothing more than a hand-wavy insult. He's a politically powerful billionaire, he has an undergraduate philosophy degree, and he likes to spout about politics and apparently the Antichrist. That's really all there is to it. The use of the term philosopher-king has no intellectual value in this context.
You would have had the same problem if you had said, "Look up Napolean. It explains a lot." The problem is that it explains nothing. Neither Plato nor Napolean helps to explain Peter Thiel, or Elon Musk for that matter. "Philosopher-king wannabe" or "Napolean wannabe" is nothing but an insult wrapped in an historical reference, with no intellectual or explanatory value, and no specific connection to Thiel.