It's not true that anxiety is the reason you stutter in front of people, since it's not a psychological issue but a neurological one. When you talk to someone, the part of your brain responsible for socialization is activated, whereas when you're alone, it isn't. When this part is activated, that's when you stutter. Obviously, anxiety can affect stuttering, but the reason you'll never recover is this: because your neural connections adapted to language are weaker than those of a normal person. And because they affect you when you speak with someone.