I believe it was mostly a chest-thumping move. "We are America, hear us roar!" The original
MCLMM moment.
The atomic bomb is an incredible PR stunt. Whether it crippled Japan or not was largely irrelevant:
It looked scary.
The bombs weren't instrumental in winning but they were instrumental in speeding the process up.
Nobody had used one before and the ability to destroy in a city in seconds is pretty persuasive.
That and you can't have a new toy and not use it, especially in war. Everybody loves toys.
Also: I forget what famous Japanese military leader said it but the quote went like,
"You cannot invade America. There would be a rifle behind every blade of grass."
I think that philosophy also applied to Japan. You would have had to kill everyone.
...
Winning a battle, whether its man on man or country on country requires three things:
Audacity (exploit the situation), violence of action (be aggressive), and momentum (be moving).
In this case, the atomic bomb exploited the situation, was incredibly aggressive, and kept us moving fast.
...
In the end it comes down to your politics and what type of percentages game you wanna play.