Miranda readings, which most people conflate with this "right to remain silent" are a bit of a bizarre invention of the Warren Court from other preexisting rights:
1. The against self-incrimination under the Fifth Amendment. 2. The right to have counsel present during interrogation, since interrogation is considered an inherently coercive environment under the Sixth Amendment.
If the goal is for the suspect to understand his or her legal rights during questioning, "You have the right to remain silent," is a very bad way to phrase Fifth and Sixth to an average Jane.
A more sane way might be an exclusionary rule that throws out all statements before the suspect has the chance to talk to an attorney.
To patch up the corner cases, you could probably import quite a few hearsay exceptions, like excited utterance.
Plus, I don't like this "know the magic words or else" type jurisprudence. "Am I free to leave?" "Am I being detained?" "Do you have a warrant?"
In everyday police interactions, a citizen should never have to assert Constitutional rights. The onus should be on the state's system to ensure they're granted.