But, see, this is what you're missing. Yes, in the context of the films, that lady saying his name is Robin means absolutely nothing. Which means that, taken on its own, that's superfluous information and is therefore not good writing. Imagine if the scene had played out exactly the same way, only the lady says, "You should use your real name more often: Alastair." You'd be shaking your head going, "What the fuck was the point of that line?! What does that have to do with him being named in Bruce's will, his character arc, his future, anything?" And that's exactly what that line would do to anybody who doesn't know what significance the word "Robin" has to anything.
"But," you say, "there's no one who wouldn't get that reference, therefore it's not as meaningless as if it was some other random name." Well, right you are. And that's my point. This is Batman from the ground up, but in this case, it is forcing you to have prior knowledge. You have to know who Robin is for that line to make the slightest bit of sense. They knew who they were writing that line for. These films don't exist in a vacuum where no other Batman mythology can't seep in through the consciousness of its audience. The name "Robin" carries a very strong connotation when used in conjunction with Batman, whether the character has been introduced or not. If there had just been a scene where Alfred tells a story from his past where he accidentally killed a bird, if they wrote it as a mockingbird, no one would think anything of it other than what's at face value. But if they called the bird a robin, even if it was just by accident, and they had no purposeful intention of making an allusion to the character, all the fans are going to assume it is, because that's how strong the connection to that word is in this franchise. Were that not the case, my theatre wouldn't have erupted with applause when that name was spoken (including myself... I was caught up in the moment and am a Robin fan). But you can't have it both ways. Either it's a completely meaningless line or it's there to serve a purpose. But it's presented so overtly that it can't possibly be both.