King+pawn vs king can only be won if the other guy makes a mistake. However, the other guy can make a mistake. The draw situations are king+knight vs king or king+bishop vs king.
The K+P vs. K scenario being discussed is not the one where the K player dies (loss, because K+P could have won). It’s the one where the K+P player dies (draw, because K could not have won).
K+N vs. K or K+B vs. K are already an automatic draw regardless of the players’ ability to make future moves, so death would not be a relevant factor there.
K+N vs. K or K+B vs. K are already an automatic draw regardless of the players’ ability to make future moves, so death would not be a relevant factor there.