Here's what much of the media have missed: regardless of whether the public-safety exception applies, the government is not, under the fairest reading of current Supreme Court law, constitutionally obligated to Mirandize Tsarnaev -- or any suspect for that matter. In the furor over the exception and the Republican senators' dubious stance, the media have conflated the issue of (a) whether or not Tsarnaev has a constitutional right to be Mirandized with the issues of (b) whether or not the public-safety exception was properly invoked and (c) whether or not Tsarnaev may be treated as an enemy combatant. The issues are distinct.
Miranda establishes that statements made by a suspect in custody in response to interrogation are not admissible against the defendant in court unless the defendant has been properly Mirandized. Reading Miranda, one would be forgiven for thinking that law-enforcement agents are required to issue the familiar warnings regardless of whether they intend to use the statements in court. The Warren Court in Miranda stated that a suspect in custody "must be warned prior to any questioning that he has the right to remain silent" and so on (emphasis added).
But as with many of the constitutional rights recognized by the Warren Court in the field of criminal procedure, the Supreme Court has chipped away at the Miranda doctrine in subsequent cases. In recent years, pluralities for the Court have clarified that the privilege against self-incrimination (the Fifth Amendment right that Miranda protects) is not violated by mere questioning; rather the right is only violated when, unwarned -- to Mirandize is, in effect, to warn -- statements are admitted at trial.
In the 2004 case United States v. Patane, a plurality for the Court stated that "deliberate failures to provide the suspect with the full panoply of warnings prescribed in Miranda" would not violate the suspect's constitutional rights or Miranda. It made the same point in the 2003 case Chavez v. Martinez. While these cases may be discounted as not holding the same precedential weight as majority decisions, they represent the best understanding of the state of the law today.
Thus, failing to Mirandize Dzhokhar Tsarnaev is not, in and of itself, a violation of his rights. The authorities are not constitutionally obligated to Mirandize Tsarnaev anyway, so long as they do not intend to admit Tsarnaev's statements at trial. What the public-safety exception does -- if and only if a court determines that the exception was properly invoked -- is render Tsarnaev's unwarned statements admissible as evidence where they otherwise would not be. And even where the public-safety exception applies, the substantive rights that Miranda protects don't disappear: due process is in effect; any coerced statements remain inadmissible; and Tsarnaev may not be denied access to an attorney if he asks for one (though the federal circuit courts have held that questioning may continue for some period of time under the public-safety exception even after the request for counsel, and statements remain admissible).