The DA feature was not chosen for any safety reason, it was accepted because it gave a second strike capability to (hopefully) fire a dud round. That was a much higher concern in the latter 1930s than any safety considerations.
I'll quote here from Capt. Edward C Crossman writing in the early 1930s on the Walther PP. His comments are reprinted in the book "P-38 The first 50 Years" by Gangarosa...
"...They embody more clever and interesting features than any other automatic pistol in the world. The double action feature solves the problem of carrying an automatic pistol chamber loaded, hammer down, perfectly safe and yet ready to go into action with the speed of a double action revolver..."
Early ads for the PP and PPK emphasize the safety of the da/sa action and the ability to bring the gun into play one handed without having to rack the slide to bring a round into the chamber.
One ad points out that with a round in the chamber and the safety on...
"In this state the pistol can be carried without the risk of an accidental discharge and yet is ready for action the moment of danger or necessity without the help of a second hand, because the marksman presses up the safety lever...and pulls the trigger as with a revolver in order to fire the first shot."
It was also emphasized that the gun was safe to carry hammer down on a loaded round with the safety off.
Second strike capability was also seen as a feature. In the PP, P and P38 though this capability was a secondary result of the da/sa advantages.
These were the features the German military wanted in a service sidearm. Gangarosa in his book and Kersten in the book "Walter-A German Legend" make this clear. They wanted a gun less expensive to make than the P08 and more reliable. Walther had developed a reliable da/sa pistol design and they wanted those features for their new gun. They got it in the P-38.
Some militaries and later law enforcement saw problems with single action pistols. Their solution was not proper training in gun handling but to build their way around it to what they considered a safer design.
There is nothing inherently unsafe in single action pistols.