Why did the Americans think it was a good idea to invite the Japanese to a neutral island and demonstrate the power of the atomic bomb?

That's an interesting question! Due to the toll in human lives, the Allied powers were desperate to end the war as quickly as possible. At the time, the Japanese military had perpetrated atrocities all across Asia, with no end in sight. I won't go into Japanese war crimes in my answer, but perhaps you may be interested in the links below:
Japanese War Crimes from the United States Inter-agency Working Group (IWG)
Statistics of Japanese Democide
Here is a A Cross-Section of Japanese War Crimes, including the Bataan Death March, The Bangka Island Massacre of Australian army nurses, the murder and cannibalism of American pilots, and similar other crimes.
The scientists who worked on the atomic bomb believed that a bomb demonstration on a deserted island could convince Japan to surrender. So, why were they opposed to immediately dropping the bombs on a Japanese city? First, some feared that the atomic bomb would fail to do enough damage and would further embolden Japan in its imperialistic ambitions. Also, the scientists feared that the use of nuclear weapons could set off an arms race and result in the threat of a future nuclear war.
So, these are the two main reasons for why some experts wanted a bomb demonstration. Despite their recommendations, the Scientific Panel that advised the United States Secretary of War made two opposing recommendations:
1) That the United States should not perform a bomb demonstration for the United Nations. Instead, it should quietly inform its allies about its intentions to use the weapon.
2) Since the panel could not propose any other way of ending the war, they concluded that there was "no acceptable alternative... [other than] direct military use" of the atomic bomb.
In the end, the atomic bombs were used, first on Hiroshima, then on Nagasaki days later.
For more, please refer to the links provided.
https://thebulletin.org/2016/08/why-the-united-states-did-not-demonstrate-the-bombs-power-ahead-of-hiroshima/

https://www.nps.gov/articles/trumanatomicbomb.htm