Adolf Hitler never did a single thing in his life because he felt obligated to do so.
Okay, sorry, a bad formulation for what I meant to be more like "Hitler felt psychologically obligated to get his piece of the action at that juncture". Not contractually so, because breach of contract was a standard military maneuver for him and several other prominent WWII leaders.
Hitler certainly wanted a war with the United States, preferably at a timepoint of his choosing; not merely the defeat of the US (perhaps to be handed to him on a magic silver platter without a war), but also the political convenience of having Roosevelt placed in the corner of being in declared war against Germany. That e.g. made it easier for him to fight the Atlantic battle, as the flow of ships to the British isles was rather efficiently disrupted by striking at the soft and unprotected American side of it.
However, let's not forget that the Leader was an opportunist beyond belief. If the interior and exterior policies of the US developed to his liking during the late 1930s, a war against the US might well become unattractive. If there was no Roosevelt there, itching to go to war against Germany as soon as another Lusitania could be invented, then Hitler's best bet in keeping Churchill starved might have been to declare war on Japan instead. Or, a US government sympathetic to the German cause might be persuaded to leave Japan in peace despite its Chinese and Indochinese exploits, giving the Japanese a better theoretical chance to hit Stalin at Hitler's behest, and thus perhaps even actually plunging Japan to this suicidal undertaking despite their natural disinterest.
The important thing here is that our "What if?" exercise begins well in advance of the actual war. Things start to change during the Great Depression, or as Spock puts it, circa 1930 (but potentially a bit earlier, as IIRC we don't actually see printed dates confirming Spock's guess). This allows for several key differences in prewar escalation and diplomatic doctrines. It even predates the Japanese expansion to the continent, allowing that to go differently.
Since Spock had recorded both timestreams and basically did a compare between them it would make sense to identify the first difference as the point of change.
But Keeler's peace movement would hardly be the first difference. It would be something nicely observable several years after the divergence point, yes. And once one observed it, one might notice that the kingpin person involved happened to be somebody from the city where McCoy and his friends were meddling with time. But other changes would no doubt take place as well, and any of them could, should and probably would lead Spock astray in his search.
(I wonder, how come Spock knows this much about Earth history in the first place? In "Space Seed", he seemed to have trouble comprehending what humans mean by Third World War...)
Timo Saloniemi