Did Jesus have the human sinful nature? No, he did not. Just as Adam did not have the "human sinful nature" UNTIL he sinned, e.g. disobedience . . . Jesus did not have the "human sinful nature" . . . nor did he acquire the "human sinful nature" . . . because he remained without sin, e.g. obedience.
Why did he die? He was the sacrificial Lamb. He gave his life so that those who believe in him could be cleansed and their sins forgiven.