There are several difficulties with this claim. The first is guessing what would have happened without the atomic bombings. Would the US have had to have invaded the home islands? It's not clear; there are many reasons to think Japan would have folded prior to November 1945, when the invasion of Kyushu was set to begin (the invasion of Honshu would have been several months later and was not even approved by Truman yet). Even if that invasion had taken place, would the casualties have been massive? This is also hard to estimate. Over the years, apologists for the atomic bombings have increasingly estimated larger and larger casualties for the invasion, both American and Japanese. But these are hypotheticals. Would the Japanese have surrendered after the invasion of Kyushu, the invasion of Manchuria, and after the naval blockade? Would they have surrendered after any one of these events? Would the combined casualties have been more than the +200,000 civilian deaths at Hiroshima and Nagasaki? I don't know. Anyone who authoritatively tries to tell you one way or the other is trying to sell you on their political agenda, not a real historical argument.

The second issue is in making assumptions about what the alternatives might be. If you get forced into the "invasion or atomic bombs" question, it's hard not to side with the atomic bombings, at least because one can imagine huge casualties for the invasion if one wants to. But there were several other alternative approaches to the end of the war on the table at the time. These include "diplomatic" options that some members of the Japanese high command were trying to pursue that would end the war without any invasion. Would any of these have worked better? I don't know. Nobody knows. But they do point to the fallacy of the "two atomic bombs on two cities in three days versus a massive invasion of the home islands that would be fought for every inch" being the only two options.

As for the leaflet question: no leaflets were dropped that warned any cities about atomic bombs prior to Hiroshima and Nagasaki being bombed. LeMay leaflets named cities other than Hiroshima and Nagasaki as the targets. As far as I can tell there were some generic leaflets dropped over Hiroshima, but they did not warn of any imminent attack. In any case, it is a false moral argument at its core: even if you do give a vague warning about bad things to happen, you are still responsible for the civilian deaths that come from it, and at no point were any actions taken in the planning of the atomic bombs that were intended to reduce civilian deaths. Indeed, every action in the planning of the use of the bombs was done with the goal of maximum urban destruction, to send a message. One can see how that would be the case in the context of the war, etc., but one shouldn't be euphemistic about it.

First, note that

Did the atomic bombings in Japan indirectly prevent the deaths of more Japanese citizens than they killed?

and

Would the war have gone on long enough to kill more lives than were lost in the atomic bombings had the bombs not been dropped?

are two quite different questions. The first only considers Japanese (civilian) deaths, and the latter non-Japanese deaths. The Japanese occupation of SE Asia led to perhaps 6 million deaths, and their occupation of parts of China to perhaps 15 million deaths. This is approximately 3 million deaths per year, or about 300,000 deaths per month. If the atomic bombings, with about 200,000 dead, finished the war 3 weeks earlier than without them, they saved lives.

Japan would probably have surrendered without the atomic bombings, but it would have taken longer, and perhaps an ivasion of the Japanese home islands. There were already starvation deaths during the war, and in the immediate post-war period. US food aid during the occupation saved many, many Japanese people from dying of starvation - perhaps the high estimates of 10,000,000 saved are exaggerated, but it is very likely that prompt surrender and US occupation and food aid saved more Japanese lives than the atomic bombings killed.

If the Japanese did not surrender without invasion (in the absence of atomic bombings), one needs to consider the likely deaths as a result of such an invasion. In the event of US-led invasion of Kyushu, the Japanese military deaths would probably have far exceeded the deaths from the atomic bombings. During such an invasion, civilians deaths would have very high; civilian and military deaths together would probably have been well over 1 million.

The maths is simple. Not even counting Japanese lives, each day the war finished earlier saved perhaps 10,000 lives. Add to that the prospect of rapidly increasing starvation deaths in Japan, and deaths during an invasion, the atomic bombings saved lives.