No, the phrase isn't meaningless. Imagine, a mother says to the boy: Put on the red shirt and blue pants. And he replies: No, I don't want to wear that. I will decide (what to wear). I am already a big boy.
I disagree. In that context, he would say 'I am a big boy now'. If it was a stage stereotype of a Jew shrugging his shoulders and saying 'I'm a big boy already' it might be more likely, but that still is not the translation of the phrase.