He's not a public intellectual and he doesn't want to be. But people treat him like one regardless, because he happened to attain a little clout as a "Rationalist".
He doesn't get to decide whether he's a public intellectual. If you publish your thoughts and ideas about issues that are of interest to the public, you're a public intellectual, whether you like it or not. Even more so if many people read what you publish, discuss it, and give it weight.