That only happens not after you have learned the piece, but MEMORIZED it, and that is another process altogether. "auto-pilot" a song that you have not learned to play without mistakes and not taken the time to memorize means playing that piece with more mistakes... and that's a slippery slope.It is interesting to see that several of you mention the "why" as a key point, and it has made me think.
What I aim for is to play a song without having to do the thinking part. Just like driving a car, I want my body to play it automatically, so that I can shift my focus from "hitting the right notes", to conveying them in the way I want to.