It'll be interesting to see what does happen. Some of these kids do seem rather smart but incredibly naive (hey, reminds me of me at the age of 19) -- what does happen when Asteroid Mining Kid runs up against cold hard reality?
Let me just clag in his whole description: John Burnham believes that the search for new resources has driven exploration, expansion, and innovation—from the discovery of the Americas to the California Gold Rush. Likewise, he believes the key to colonizing space is to make it possible to extract valuable minerals from asteroids, comets, and other planetary bodies. John plans on using his fellowship to develop space industry technologies to solve the problem of extraterrestrial resource extraction.
Really, what can you do to advance space industry technologies under these circumstances? You're a teenager. You have a hundred thousand dollars (aka what Elon Musk spends every four seconds). You know far less about space technology than 99% of the people who work in the industry. And you're very smart, but a lot of people in the rocket industry are very smart. What do you do? Go intern for Elon Musk?
I've worked in the space industry, and sadly, like in any tech industry, it is actually likely an incredibly intelligent kid would know more about the math and science of space technology than the vast majority of those in the industry. Sure, they wouldn't know the project-specific details, but those would be easy to get caught up on (for someone of that caliber). This is why the private space ventures are doing so well: they concentrate the real experts and cut away the bureaucracy. Also, keep in mind the space industry hasn't necessarily attracted the majority of the best and brightest for a while now; think of computers, biotech, and finance.
That said, lack of power in aforementioned bureaucracy could be a major issue.
Actually in that position, interning for Elon Musk sounds like a great idea, followed by using that $100k to fund living expenses in college and grad school, with the bonus of having a streamlined academic plan and semi-guaranteed dream job when you graduate.
I have a friend at Stanford who got a $100k fellowship for graduate school, and is interning at Tesla this summer. He's doing the exact same thing as you're proposing Asteroid Miner Kid would do, but through the completely traditional route. Seems like a waste if AMK does the same thing as my brilliant but more conventional friend.
But it's because of those lofty ideas that we have so many great things now. Just because someone does not posses the knowledge of his/her peers regarding a certain field, does not necessarily make them unqualified to work in that field. On the contrary, they are not as bound by all the previous work done in the field, and they may very well come up with something ingenious.