The Barbarossa aspect of this could be argued two ways. Would a six week head start been to Germany’s benefit? The attack caught the Soviet Union completely off guard, and resulted in the rapid encirclement and surrender of millions of Soviet soldiers. Germany achieved a 10:1 exchange rate during Barbarossa.
The problem with starting all that six weeks earlier was that spring rains were worse than usual that year, and persisted later into the spring than expected. The effect of surprise would, at least initially, have been largely counteracted by muddy roads. That would have given the Red Army the chance to get the worst of the surprise over with, and at least begin the process of getting its legs back underneath itself, at a time when Germany could not take maximum advantage of the situation. It’s possible that they could have accelerated the beginning of Barbarossa by a week or two without running into the muddy roads problem. But not six weeks.
A second point to bear in mind is the value of not having the British military on the European mainland. Italy might have eventually been able to defeat the Greek army. But as Britain sent more and more reinforcements to Greece, the combined Anglo-Greek force would have quickly become too tough a nut for Italy to crack. Britain could draw troops not just from its home islands, but also from its enormous colonial empire. The last thing the Axis needed was for Egyptian or Indian soldiers to gain a stronghold in the Balkans. If that force grew strong enough Germany might eventually have been forced into a two front land war, even without American involvement. There was also–as I mentioned earlier–an absolute necessity to keep British bombers from bombing Romanian oilfields. Without that oil the German war machine could not function.
I don’t think that the invasion of Greece was responsible for the full six week delay in Barbarossa. If Hitler had gone for Greece only, maybe it would only have been a three to four week delay. That’s acceptable due to the late spring rains. Unfortunately for the Axis, the government of Yugoslavia was overthrown in a military coup. As Hoover pointed out in his book, FDR encouraged that coup by falsely promising members of the Yugoslav military that, if they overthrew their own government, the United States would provide military aid against German retaliation. With perfect foresight, the Germans would have allowed Yugoslavia to remain neutral, instead of pressuring it to join the Axis.
But let’s say their foresight wasn’t quite as good as that decision would have implied. The next-best decision would have been to launch Barbarossa even while mopping up opposition in Yugoslavia and/or Greece.