PURPLE's decryption was largely irrelevant to Pearl Harbor; it was only used for diplomatic messages, while the attack was a military operation without involvement from the diplomatic arm. The Japanese ambassador and staff were continuing negotiations over the embargo and occupation of French Indochina right up to December 6. The decoding of the critical message breaking off negotiations was only completed December 7, at which point it was accurately determined to mean that an attack was imminent. The warning, however, was sent out too late to be of any use; it didn't even arrive until after the attack had taken place.
I'm also curious about the assertion that the Kido Butai was detected by radio traffic. The Japanese plan called for the use of semaphores and light signals to transmit signals between ships, going so far as to physically disable the ships' radios to prevent any incidental signals from leaking out by accident, a precaution corroborated and confirmed to have been in place by their own after-action report. Given the perceived success of the operation, it seems unlikely they would have systemically lied throughout the highest levels to turn in a false report. That said, we do know that IJN signals in general spiked significantly when the Kido Butai set sail due to the operational orders sent out to other task forces, but this was no immediate indication of an attack on Pearl Harbor. There was significant, and admittedly naive expectation that the primary targets of any sneak attack would be the Philippines due to the foreknowledge that the Japanese would enter into any such war with the aim of seizing the DEI, and this assumption seemed to inform most preparations in the Pacific theater. Unfortunately, MacArthur was not ready for the attack that did eventually come even with forewarning of the events of Pearl Harbor; with ten hours of warning, there was still this persistent underestimation of the Japanese which, combined with local lack of preparedness, resulted in the Far East Air Force being devastated on the ground.
The thing that clinches it for me, however, is Roosevelt himself. He was very unabashed in his willingness to prosecute hostilities with Nazi Germany. As such, it seems unlikely that he would intentionally invite an attack on Pearl Harbor that would redirect American attention away from Europe and towards Japan. Remember that the outbreak of hostilities between Germany and the US was a consequence of actions by Germany: to wit, Hitler's declaration of war in support of Japan. At the time, the only precedent established in the alliance between Japan and Germany had been the exact opposite: Japan had refused to invade the USSR with Germany mid-1941, had declined to immediately invade the Dutch or British Far East possessions in 1939 or 1940, and had instead attacked French Indochina, a colony which had sided with Vichy France and thus was in a state of armistice with Germany. As such, the reasonable expectation would be for Hitler to cheer for American distraction and quietly provide Japan with at most only moral support, just as Japan had done to him. If we accept the postulate that Roosevelt somehow played the chessmaster to cause a war with Japan, as opposed to simply and catastrophically misjudging their willingness to resist the cessation of war with China in a status quo ante, then the only reason it worked was because Hitler, against all reason, chose to allow it to work. If Hitler had not declared war, the US would have been stuck with a Japan-first strategy and would likely have been forced to pull back forces from the Atlantic, where the policy of "proactive defense" was beginning to bear fruit.