Probably late to the party, but:
I believe that it both happened, and did not happen. I think the truth is far more complex than what one side (6 trillion died!) and the other (they didn't die! but they deserved it!) believe and hold as gospel. My view is that, initially, their plan was to simply intern them in concentration camps - akin to the internment of Japanese-Americans contemporaneously in the United States and the Boers in South Africa only forty years prior - and keep them there until some kind of deportation was arranged. There are clear efforts of the German Government attempting some form of this, with the Haavara Agreement in 1933 and the various schemes like the Madagaskar Plan, but for one reason or another this fell through.
Following this, the actions of the Einsatzgruppen in occupied Russia speak more on general violence directed in occupied territories, which is commonplace throughout history, but clearly with a directed slant against the Jewry. There was also clear collaboration with this with the local citizenry, as in cases like Babi Yar. In this "period," the MO is a clear line from other actions taken by armies throughout history: open massacres and clearing of villages and areas. But was this a directly sanctioned and ordered action from the high command, or spontaneous violence? I lean towards the latter, of spontaneous acts of violence orchestrated by unit commanders. These tend to be how massacres and war crimes happen, across all nations. Not necessarily sponsored by the state apparatus, but undertaken by those in command on the ground.
But I guess this thread is more clearly about the camps. My view is that there was no clear action taken to specifically murder all of them, as in Hitler directly ordered them to start killing them all. I think it was one action that snowballed into a bigger thing, undertaken sporadically and carelessly and without real oversight or reasoning. The Final Solution, as it's said to have been conducted, seems to have originated between Himmler and Heydrich, probably spurned on by Himmler. If we trust it, and we draw the line from there, it seems that the actual actions undertaken under that order are vaguely implemented by various unit commanders in different ways.
I would say, however, most of the deaths that happened at those camps were by and large due to willful neglect by the Germans, who had neither the reason nor the capability to feed, clothe, or medicate any of the prisoners when by that point, they barely had the capability to reliably do that for their home front or their Army.