Much of what you state is basically true, but bear in mind that other options, that didn't involve nuclear terror, would still have worked, and weren't used for reasons a bit more cynical than exasperation at Japanese truculence, or mere ignorance of radiation's insidious effects (which wasn't as deep as one might think, even in 1945)..
Even without concretely knowing just how long the Japanese government would attempt to hold out in the face of conventionally inevitable defeat, political and military planners in the U.S. could have -and knew they could have- simply sealed off the Japanese Home Islands and starved them into submission while continuing to bomb their cities to ashes with ordinary bombs.
Well before July of 1945, Japan was essentially defenseless against aerial bombardment and its cities aside from the famous cases of Nagasaki and Hiroshima had already been absolutely devastated by repeat massive bomber raids that the feeble Japanese air defense systems at the time could do next to nothing against. Many, many more people had already died in these firestorm raids than were eventually killed by Little Boy and Fat Man.
What's more, the Japanese Navy was by that point basically in scattered, useless tatters while U.S. naval and aerial might was so total, so all-encompassing, that the severely import-dependent Japanese homeland could have been brought completely to its knees in misery, to the point of eventually just succumbing, without having to send so much as a single American soldier onto the the main islands.
It would have been a longer and less "glorious" way of leveraging a surrender than one big military invasion by all those shiny war tools and soldiers you now have geared up in such abundance to fight and finish the whole mess, but it indisputably would have worked; even nationalist fanaticism short circuits when literal mass starvation becomes completely inescapable, ravages every corner of society, and the most basic essentials of modern civilization are gone, while every single one of your cities is now ashes and rubble. All of these horrors could have been achieved without invasion.
The United States knew all this. However, as mentioned above, its Marine Corps and vast navy were already at maximum deployment, so the political sacrifice of a decisive invasion was the alternative to which the atomic bombs were compared. Secondly, the U.S leadership now had the bomb, after all that cost and effort, and felt a pressing need to showcase its live military use to themselves, to the world, and very crucially, to Stalin. By that point, in July and August, he was no longer quite the friendly "Uncle Joe" that American wartime propaganda had painted him as for the sake of political convenience, while the Nazis were still in the picture.
Personally, i'm glad the bombs were used then, in that context, while their destructive power was still s relatively weak. It showed the world the monstrosity of atomic bombing used against an actual society, while the consequences for using it live were much less than they would have been if delayed to just a decade later. If at that future date, someone had decided to finally get an itchy trigger finger for the first time, they'd have been doing it with many more, far more powerful bombs, claiming many more lives..