The Death Korps are very much themed by the dour and horrific nature of the war that inspired them. They're supposed to evoke images of glassy eyed, dead staring trench veterans from the Somme, or the poor propaganda fed fools who charged over the lip of trenches into machine gun fire. They're the men (and women in 40k, though not in the real life version) who watched their friends sawn in half by bullets, their guts scattered across no-mans land by mines, their toes gnawed off by rats and their own severed limbs lying in pools of blood. They are the human horror of chemical warfare, of summary executions for shell shock, of thousands of lives thrown away in pointless war in physical form.
Where the Death Guard and Iron Warriors represent these things in the manner of demons and war gods, beings who revel in bloody and horrific war, the Death Korps represent them in the human impact side of things. As a people they are broken, traumatized and indoctrinated to the point that the horrors of wars that surpass the foulest trench of the Great War is something that hangs about them like a shroud even in peacetime, and they can no longer care.
In a manner of speaking they're a nod to the mutilated and traumatized veterans of WWI, for many of whom that war never really ended. It stayed with them, in the form of haunting memories of shellfire, the sight of their friends dying, the phantom pain of lost limbs. Kriegsmen are much the same, except they come to a certain extent pre-traumatized.
They're also a nod to the unrealistic expectations of discipline and cleanliness expected of the soldiers in the trenches. A lot of punishments were doled out during that war for people failing to maintain a near parade standard in the middle of miles of cloying mud, bug infestations and shrapnel. Though the obsession with maintaining gear is also a possible sign of PTSD as I recall.
Kriegsmen represent the ideal soldier the generals of WWI wanted, which is to say disciplined, obedient, presentable and unquestioning meat puppets who'll accept the horrors of trench warfare without question.