The argument for the first bomb was very different from the argument for the second, though. For the first, it demonstrated 2 things that would be hard to do without a strike on a city. 1, the effect, and 2, the will to use it on a city. Shock and awe, and commitment, basically.
But the second one? Those points were proven, its purpose was to show it wasn’t a one off and that could have been done with an offshore drop, or on a less inhabited piece of land.
TBH there’s room for argument on the first and lots of good arguments on both sides. I’ve never seen anything that convinced me Nagasaki (or rather Kokura, which was supposed to be the target) was justified.
As far as the big picture- it’s complicated. General LeMay believed it was a war crime even as he gave the order, Szillard believed it was a war crime while he helped develop the bomb. War isn’t tidy, it’s awful, and what people seem not to want to do, is to accept that it could be simultaneously state terrorism, a war crime, and yet justifiable. Those men went into it with eyes wide open, it’s kind of weird that a lot of people nowadays seem unwilling. Maybe it’s a reality of war thing, the more divorced you are from the reality the harder it is to believe the extremes it takes you to.