Sure you are right. If you trained under ICAO rules, you are correct. But can someone learn all of the FAA Private PTS maneuvers to standard, meet the night and simulated requirements (not every country requires the same thing), and other requirements in 3 hours? Nope. I have had to do this 5 times in the past year with students who got their PPL certificates in various locales around the world.
Moral of the story is exactly as you say: train in the US, then take a flying holiday to SA with the money you save.