Peru isn't anywhere near the poorest country in the world. In Somalia and Burundi, that's about the per capita GDP for a year. Unsurprisingly, there isn't much of an obesity crisis there. If people just ate less of the more energy dense foods, they wouldn't be fat. Obviously they'd have to adjust their diets to the new staples, but a more energy-dense staple needs to be served in smaller quantities. Obesity is a simple equation of eating more energy in food than you use. Unfortunately, lots of people around the world struggle with this.
You can't say that people have no choice to be fat because they have potatoes and rice instead of quinoa. In many parts of the world, those have been the staples for centuries.