I'm not so sure about that, not all societies that can "have all the food" they want are fat, for example France.
I'm a French person who's been living in the US for about 5 years now. My main realization regarding the way food is treated in each country is that in France, the relationship with food is conditioned culturally very strongly since the moment you're born. When you grow up as a French person in a culturally French family, you grow up in an environment where you constantly hear about what foods go and don't go with one another, how you should eat, etc. You hear this through your own family, through meals at schools, through meals you share when you go to your friend's house, and so on. Little things like "you're supposed to eat bread with your salad", "we've already had meat for lunch let's not have it for dinner", "don't eat with your hands", etc. For the first 18 years of your life, you hear these things on an everyday basis to the point that when you start living on your own, they're part of your subconscious. If you ate pizza on Monday, you're not going to eat it again on Tuesday because you'll hear your grandma's voice in your head that says "You already had pizza last night, eat some vegetables tonight". When you have dinner with your friends in college, you're likely to still cook something fairly healthy because most people share that upbringing with you. And then when you raise a family of your own, you yourself repeat these things and the cycle continues.
This culture has evolved and been passed from generation to generation over centuries, and the result is that the average French person has a fairly strong instinct for "eating healthily" because of this, even though they're not compulsively counting calories at every meal. Their cultural baggage means they can just cook according to their gut, and their meals are likely to be balanced over time. The same is probably true for other countries with strong food cultures such as other mediterranean countries, Japan/South Korea, and so on.
Such a culture does not exist to such an extent in the US. I've found the contrast to be pretty stark - when I eat with French friends, there's an entire set of manners and habits that are never expressed but mutually understood, which does not happen when I eat with most American friends. As a teenager, I lived in the US with a host family for a summer who never ate together - different family members would just pick and choose from the fridge at different times of the day as they saw fit, with no regards for whether their food intake was balanced or not; then maybe once or twice a week, they'd have a barbecue all together. This would be completely inconceivable in a French family.
That being said, France is changing, families are becoming more and more multicultural, and things like fast food restaurants are changing norms.
In russia, where climate is hard, earth is not very fertile and one unsuccessful harvest could mean country-scaled global hunger, food culture is more about eating while you can, conservate food while it's fresh. With current excess of food there are many fat men and women out there.
I believe that all countries will develop food culture very soon, it's already happening around. Dietology is a popular topic.