Boy, we American are such pompous dicks, aren't we?
In all seriousness, the problem you describe is very complicated and it branches from the tidal wave of propaganda that was out during WWII. You see, after WWI and for many other reasons, Americans actually wanted nothing to do with wars. WWI was a massive time and money sink that left America with no benefits, since the treaty to the League of Nations was shot down by Congress. Americans wanted a neutrality policy (or the "Fuck off" policy, as I call it), and for the longest time FDR satiated those desired with two Neutrality Acts. He wanted to stay out of the war as much as everyone else.
But then the war grew to large, and after Pearl Harbor was attacked, FDR felt he had no choice in entering the war. Most Americans wanted war after Pearl Harbor, but there were still large amounts of people that did not. Plus, FDR and the army had to get people to donate food, clothes, etc to benefit the war effort. How could they do that? Propaganda. Look up "WWII Propaganda" right now on Google. Look at the imagery. The images that prey on people's fears. Soon, WWII became about fighting the fears posted in those posters. Then, WWII (especially the support on the home front) became about supporting one's country, and if you didn't, you were just as bad as a Nazi. It created a wave of nationalism, which was actually a cause of WWI. The policy worked, clearly, and its effect satiate into today's society. Those who do not know any better think what FDR wanted them to believe: America is a badass country that won the war single handedly, and if you don't believe that, you aren't an American. Most people who know the horrors of WWII and what actually happened know that's not true, but not everyone learns about WWII in detail. It's sad,but true.
As with the case of Hiroshima, nobody in America is "proud" of that. Oh sure, we may joke about it (I don't know why we do, but we do), but we aren't proud. Anybody that is proud is a fool and an idiot, and will be treated as such. But once again, the nuclear bomb is a complicated situation. Now, there have been theories of Japan considering surrender not because of the bomb, but because the Russians declared war on them. But Russia declared war AFTER the bomb dropped. That may be true, but Truman did not consider Russia declaring war on Japan. He saw he had two options: the bomb or another D-Day operation, only this time on the shores of Japan instead of Normandy. Hundreds, even thousands of Americans could have lost their lives. Truman chose the bomb. Was racism a factor? Maybe. Truman wasn't the most kind and most considerate President we had, but he decided to choose American lives over Japanese lives. Was he proud? I doubt it, but he knew he had to make a decision. Also, I am not sure why he decided to drop a bomb on civilian cities. Perhaps someone could look it up, but I am sure someone had a reason.
TL;DR: Americans are not proud of WWII and the nuclear bomb; our supposed "disillusion" stems from the series of propaganda used by FDR during that time to convince a war weary public that WWII was something that is heroic and something to be fighting for. That thinking unfortunately has leaked its way into today's society for those who do not know the horrors of war.