Regarding the notion of Auftragstaktik and Battle drills:
IMHO the essence of Auftrakstaktik is constituted of the idea that an order is a description of what to achieve but not how to achieve it - leaving it to the subordinate who has better judgement of the situation he encounters to understand the superior's intentions and how to implement it in a given situation.
This implies that for instance am enlisted man had to be able to take over as a squad leader, a squad leader was also educated as a platoon leader, etc...

Historically, the necessity to introduce such concepts in Germany lies in the unique and unfavorable strategic location of Germany which was "encircled" by other powers. Therefore, German armies throughout the history were often outnumbered and short of supply. So creativity was a necessary ingredient to compensate for the odds. I believe it was first mentioned in the liberation wars against the French occupants under the rule of Napoleon.

However, I also believe that the Wehrmacht did exercise something like battle drills (e.g. ballistic machine gun fire together with advances requires frequent rehearsals) - but they were not to be followed blindly - and flexible tactics were encouraged (although on a strategic level the high command always had to struggle with often insane orders given by Hitler).

Notice, that German armies have been draft based for hundreds of years.

Interestingly, as media becomes omnipresent on the battle field, the army becomes more professionalized, due to the technological improvements in communication technologies, and politicians play General and Generals have to cover their asses, Auftragstaktik slowly starts to erode in the modern German
forces...