South America was the most "neutral" part of the world, on a globe that was overwhelmingly pro-Ally.
Germany, Italy, and Japan were all occupied by the Allies, so they were "out." Moreover, nearly all European nations had suffered from German occupation. Spain and Portugal were important exceptions, and within Europe, might have been the "best" refuge. But there were others abroad further away from British and American power.
Africa (up to the late 1950s) was mostly colonized by Allied nations, Britain or France, or possibly Italy (which had been "turned"). Most of Asia had been colonized by Japan or threatened with colonization, so little sympathy there. One possible exception was Thailand, but that was not a destination for Germans of 1945 (maybe more modern tourists).
Most of Latin America was "officially" pro U.S. But the further south you went, the less "pro." It is noteworthy that it was the Central American and Caribbean nations that signed on to the Atlantic Charter on January 1, 1942, forming the core of the UN, with South American countries joining the UN later, and Argentina joining "last" (prior to the 1945 launch). Among South American countries, perhaps Colombia, Venezuela, Ecuador and Peru were most threatened by the Germans and Japanese. In countries like Chile, sentiment was perhaps 40-60 Axis-Allies, and in Argentina and Paraguay,* probably 50-50 or "better" (in favor of the Axis). "Marching Orders"]1 tells of an unsuccessful Axis attempt to bribe members of the Chilean legislature to support them. These were also among the strongest South American nations as well as being the most "independent" of the United States.
Governments of countries like Argentine and Paraguay would not welcome the label of being "pro" Nazi. However, it was easy for them to turn a blind eye to immigrants that other countries might deem undesirable. For them to "do nothing" was to be more "welcoming" than most other parts of the world.
*Paraguay had a German-trained military that had won the Chaco war against Bolivia in the 1930s.