TL;DR, it was becoming apparent that Japan was not going to surrender, and were willing to fight to the last man standing. The nuclear bomb drops were to demonstrate to the Japanese government that they were clearly outmatched by the end of the war.
It worked.
I believe there is an opinion of some that part of the motivation was to prevent the Soviet Union from moving in after Japan's defeat, like they did in Europe.
This is often asserted as if it were a shocking revelation and a moral indictment of the decision.
However, it seems to me both a plausible motivation (I haven't bothered to read up on the evidence or don't remember if I have) and potentially morally defensible, particularly in hindsight.
https://foreignpolicy.com/2013/05/30/the-bomb-didnt-beat-jap...
Note that it's in the interests of both countries to promote the Bomb narrative - America to justify the financial and moral cost, and Japan to justify the loss of the war (a black-swan wonder-weapon being a less shameful reason for defeat than military incompetence). But in truth, Japan's cities had already been firebombed to smithereens - in that context the Bomb wasn't really all that shocking.
If you think the US only burned two Japanese cities, you should look at the history of the US's bombing campaign. Firebombs were used a lot.
As for "taking the head honchos to the tests", let me propose a few counterfactuals:
1. Japanese leadership sees bomb, changes air defense priorities to target small groups of planes and not mass attacks. Enola Gay is shot down. Invasion happens starting in October; 500k+ die at minimum.
2. Japanese leadership thinks the US is soft and lacks will to victory, holds out even after bombs are dropped. Invasion happens starting in October; 500k+ die at minimum.
3. Japanese realize this isn't really worse than the existing mass firebombs, distributes more production and population away from cities. Invasion happens starting in October; 500k+ die at minimum.
Considering how close the surrender seems to have been in our timeline (personal decision of the Emperor, overriding advisors and military leadership) I would not assume that a softer path would work.
2. If they do not surrender after demo, then drop on Japan. Invasion would never happen as U.S. was ramping up nuclear production: https://www.nationalgeographic.com/history/history-magazine/...
3. Where would Japan move all those people and production? Why did they not move them given the existing firebombing? https://www.airspacemag.com/daily-planet/deadliest-air-raid-...
Japan was confined to a fossil fuel poor mountainous island at the end of the war. Japan could only prepare for a last ditch defense of the home.
Also, the maneuvers that a B-29 needed to do to escape the blast of one of those weapons was really extreme - and Japanese engineers would likely be able to figure this out too. Those maneuvers mean that you can't fly in close formation. (Or really any sort of formation with separations not measured in miles)
2. "But we should have tried more!" is always something that can be said. I'm really not sure this changes the overall moral calculus.
3. You're linking to something saying "other US bombing raids were worse than the nukes" and using it to argue that the nukes weren't necessary. I can't really understand this either. (I'm not sure that Japan could have actually retooled to move more productive capacity out of the cities than they were already doing. But when you're trying for a psychological outcome - surrender - "we knew this was coming and planned for it" is a lot easier to handle than "what the hell just happened")
Finally, yes - Japan was confined to a series of islands with little besides coal and people. They were never going to win, not so long as the US had the will to continue the fight. But surrendering 6 months later would mean huge casualties for the invaders, massive losses for the defenders, and unimaginable horrors for the civilian population. We should all be happy for the millions of people that didn't starve to death, and for the millions who didn't have to walk their children off of cliffs or have the family hug a grenade.
Before we thank ourselves for all the lives we "saved", we should remember all of those that died in the name of an unnecessary exercise in human greed and cruelty.