For heaven's sake. Assuming that they have arrested the correct person, he and his brother had a stack of tactical explosives that they were throwing around with abandon, as well as having bombed a major public event with loss of life and a huge level of injury.
Of course the public (via its designated law enforcement agents) has a legitimate interest in establishing whether all possible explosive devices have been found and neutralized.
Bear in mind that for his trial (if he doesn't plead guilty) they'll have victim witness testimony from a person who says he saw the suspect perform a bag drop shortly before the bag exploded and blew his legs off; abundant photographic evidence, with probably more to come; whatever forensic evidence they found at the brothers' shared home; and all sorts of other information gathered in the days before he was found and captured. There'll be camera footage from the 7-11 that they held up, testimony from the person they carjacked (to whom they apparently admitted that they were the bombers), forensic evidence from the car, vast amounts of ballistics evidence...I could go on and on.
There is no way that his conviction is going to be pendant on what he says before being mirandized, and the public safety exception is about as justified here as it could ever possibly be - far more so than that in Quarles. That was a guy who allegedly had had a gun and was wearing an empty holster when caught. True, abandoned guns can be dangerous. A bumper variety pack of bombs, which it appears this suspect had and employed on multiple occasions in the last few days, presents a substantially greater risk to public safety.