 Why does the Brain select Jery to hold the amnesty?
So when the Chief of World Security told him that he had been selected as the answer to the Solar System's greatest mystery, Jery assumed that it was because of his mental agility. For Jery had become a walking bomb, and when he set himself off, it would be the end of the whole puzzle of THE SECRET MARTIANS—with Jery as the first to go! Because the speaker was Philip Baxter, Chief of Interplanetary Security, second only to the World President in power, and not even that in matters of security.
 Because of his ability to parse situations.
 To prevent someone like Baxter wielding and misusing it.  Because he's not really qualified, making him a wildcard.  Because of his ability to parse situations.  Because he will responsibly wield the amnesty.