> Doesn't calorie intake directly correlate with gaining weight?

No, calories surplus does. Poor nutritional balance can lead to low energy and make maintaining activity difficult (leading to a calorie surplus where with a more balanced diet the same person would act in a way which would produce calorie balance or deficit), and trigger a desire to eat driven by the body seeking to address deficits in particular nutrients even with a calorie surplus.

> I'm not intending to downplay the issues here, but I find it hard to follow the disconnect of people being overweight not having enough food to eat.

Nutrition is multidimensional and more than just calorie intake -- that seems to be the thing that you are missing.