Your "actual study" link is corrupt; it looks like Hubski added a couple slashes around the (19). I'll delete mine if you fix yours.

The idea that overly-processed foods drive weight gain isn't a new one. It's pretty much the basis for the glycemic index, which mainstream nutritionists have been cautioning against for decades. They have a fair point - the glycemic index was developed to help diabetics naturally manage their blood sugar, not help fat people diet - but fundamentally, it's an argument that our ability to break down food into energy affects satiety.

This study looks a little thin to me - they're showing a 20% swing in caloric intake but they're ignoring the trendline downwards in caloric consumption over time for the highly processed diet (and a 300kcal/day decrease to boot). They've also got variances of more than 10% in protein, carbohydrates and fiber between their high processed and low processed diet. They appear to have captured initial conditions but not so much long-term effects.

That said, it's not controversial to say that processed foods have more easy calories and more easy calories lead to weight gain. It just takes anywhere from three weeks to nine months to cement a new behavior. Changing someone's diet for two weeks and expecting to see routine behavior out of it is optimistic.