I’ve got a cousin who is a state’s attorney. I asked him this once. He said (IIRC) it’s basically because the cops belong to the side that will be charging you, not building your defense. Things you say might be used by the defense, but the cops have no way of anticipating what the defense will do, and cannot promise that if you tell a story about your innocence, it will be used at all. The defense may decide the prosecution’s case is so weak that no evidence will be presented. Or the defense may decide on some other approach entirely. If you choose to talk, anticipating that what you say will eventually help your defense, because the warning implied that, then essentially, you have been lied to to procure your story.
However, since the burden of proof is on the prosecution, and the prosecution must present a case, the cops can pretty safely promise that what you say will be used by the prosecution.
Even if you are as innocent as a baby, you should say “I want a lawyer,” and then shut up.
That’s advice from a prosecutor.