The question of demonstration was only briefly taken up at the higher levels (no higher than the Secretary of War), and ultimately given to several scientists to discuss. The Scientific Panel of the Interim Committee (Arthur Compton, Enrico Fermi, J. Robert Oppenheimer, and Ernest O. Lawrence) concluded in June 1945 that:

"we can propose no technical demonstration likely to bring an end to the war; we see no acceptable alternative to direct military use"

Now one can ask why they thought that. Or whether they (four physicists) were really qualified to answer a question about military tactics, national psychology, and diplomatic questions. (Especially before they actually had even seen what a nuclear bomb looked like — this was a month before the Trinity test.) Or why such a seemingly important decision was left pretty much to these four people, all four of which had their reputations and careers tied up with the "success" of the project (all were "fathers of the bomb").

The scientists in particular seemed to want to make the first use of the atomic bomb as terrible as possible, because they had in mind the idea that future nuclear weapons would be even worse. Their goal, and this was quite explicit with Oppenheimer, was that if nuclear war looked bad then the world might get together after WWII and put an end to nuclear weapons production (via what was known as "international control" measures, i.e. the UN prohibiting nuclear weapons and enforcing that prohibition). It didn't work out that way, of course, but this was on the minds of many of the scientists working on the bomb.

The military, for its part, never thought a demonstration would be worthwhile because they had spent an awful lot of money on the bombs, and they had very few of them. (Although this can be exaggerated — they only had two ready to use in early August 1945, but their production line would have produced another by mid-August, and three more in September. Which means they would have had plenty of atomic bombs on hand before the planned invasion in November.) The goal of bombs was to use them in war. They saw them as military weapons. This military/political distinction was still being hammered out, but the military were primarily running the show.

Lastly, the predominating concern was never to save the lives of Japanese civilians, but to make a "shock" in the Japanese high command. What better way to "shock" than to destroy a city? Such is part of the logic.

I have written some more on this at length here:

• To demonstrate, or not to demonstrate? (which includes the full Scientific Panel report if you are interested in their reasoning)

• Did Lawrence doubt the bomb? (about the only member of the Scientific Panel who apparently put up any resistance)

• Were there alternatives to the atomic bombings? (in general about the "choices" made regarding the atomic bombings)

On the military/political distinction, Michael Gordin's Five Days in August: How World War II Became a Nuclear War (Princeton University Press, 2008), is all about how the understanding of the "nature" of the atomic bomb (e.g., is it just a scaled-up firebombing attack, or something "special"?) was getting worked out at the time (and immediately after the war).