Yes, but there weren't enough of them. So the orders also went out to Waffen-SS battalions that were a little less inclined to engage in summary executions.
Conspiracy, a movie about the Wansee Conference illustrates the point. An SS-Major named Rudolf Lange complains about his men having to participate in mass shootings in Riga. (Actually Lange's battalion stood by while Latvian civilians did the dirty work. But it impacted his men negatively and he wanted to make that point to the Boss, Reinhard Heydrich.)
My point is that there is always a shortage of savages & mentally-ill to fill the "Einsatzgruppen". Soldiers will not do this work. If some do, it's incredibly hard to keep them at it.
Incidentally this is one of the reasons that the Nazi's turned to industrial methods that were the hallmark of the "Final Solution". There needed to be some 'distance' involved between the perpetrators and the victims.
There was enough of them to slaughter people, and create horror and misery. Effing savages...the thought of helpless people being massacred just enrages me.
One of the crowning moments of the early war was when Czech Patriots exterminated Reinhardt Heydrich. Beautiful moment of retribution. They wasted that jackbooted thug SOB right in broad daylight.
And then Hitler showed what government retribution is REALLY capable of by erasing the town of Lidice off the map.
Good that you know your history, brother...wish more people did.