Why isn't George justified in killing Lennie?

It's possible to say that George isn't justified in killing Lennie simply because it is wrong to take someone else's life. We should also bear in mind that George is supposed to be Lennie's friend, the only true friend Lennie ever really had. Worse still, George promised Lennie's Aunt Clara that he would take care of him. By killing Lennie, however noble George's intentions may be, he is clearly breaking that promise.
Also, no one has the right to take the law into their own hands. Just imagine what would happen if everyone did that. There would be complete chaos and disorder. And not only is George acting the part of lawman in his shooting of Lennie, he is also playing God. George's act of killing Lennie is presented as some sort of mercy killing, preventing Lennie from suffering a much worse fate—either at the hands of Curley or the criminal justice system. But there is something rather disturbing about this way of thinking, particularly when it is applied to people like Lennie who are clearly mentally challenged. It seems to suggest that those deemed normal by society somehow have the right to determine whether someone with mental health issues should live or die.