They dropped two nuclear bombs on Japan to make the Soviet Union's invasion of Hokkaido unnecessary and as a performative act to horrify the world in to subjugation.
The whole 'saved more lives than it cost' is a bad argument unless the plan was to slaughter a hundred thousand civilians on landing. Note the Nazi invasion of France cost 60,000 lives in civilian resistance...this was at the beginning of the war.
Japan would have surrendered on the first day of invasion.
Yeah, for real, people should read up on what an invasion would have involved.
Literal school children were being armed with spears and grenades for suicide attacks on Americans. Swarms of suicide boats were being prepared in addition to Kamikaze planes. Look at how many civilians committed suicide during the Invasion of Okinawa to get an idea of how bad it would have been.
On the flip side, the American invasion plan was to gas the shit out of the invasion beaches and basically pound any resistance point to dust with naval gunfire. None of this sounds like a recipe for low civilian casualties.