He didn't seem like a Nazi though, just ruthless because the war made him so.
What I didn't like was that throughout the entire cat and mouse game against Vasili, König outfoxed him and his partners on numerous times. Then at the very end the Major discards the discipline that had up to that point given him the edge and gets impatient. And he gets killed for it.
Agreed - Konig seems to get impatient and overconfident at the end. I cringed at the relatively huge noise he made moving the piece of "scrap metal"/debris when he exited his hideout. Nothing I can think of explains this - he was not suicidal, and he surely was not having second thoughts about having SPOILERS
... killed the little Russian kid. Nor was he acting war-weary - he was told to exit Stalingrad, but he refused, because he knew that Vasili must still be alive, because "I have not killed him". His energies and purpose have not flagged: he is still a devoted killer at the end, which makes his easy targeting by Vasili very difficult to accept.
Konig was ordered to leave the city the following day. He was then asked to surrender his dog tags so that his remains could not be positively identified should he go out hunting again and be killed. It’s not entirely clear to me that Konig didn’t intend to follow his orders and depart after making one last attempt to find and kill Vasili. If this is the right interpretation, then Konig did have some urgency which, perhaps, would explain why he discarded his usual caution.