Given this evidence, one way to reduce weight gain would simply be to switch from higher processed foods to lower processed foods. This would be a healthier diet in many regards, as an industrialized diet has many flaws of which weight gain is only one. It's a relatively straightforward change because you can actually switch one item at a time and find out which exchanges work for you or which are too difficult to sustain. Frex, switch a snack from potato chips to fruit or mixed nuts. Switch from flavored instant potatoes to baked potatoes with butter. Switch from hot dogs to pork chops, or even a meatless protein like beans. This probably won't fix everything, but probably will improve matters -- and it's easier to sustain gradual changes than big fast ones. Of course, weight is a complex thing, but it's pretty clear that the standard American diet is not helping.
The study noted a few limitations, like that it takes more time and money to prepare unprocessed foods, and not everyone has access. It also takes tools and skills -- you need a working kitchen, pots and measuring cups, cookbooks, knowing how to cook, and so on. These are not advantages which everyone enjoys, and that is a huge reason for obesity among disadvantaged people.
However, some of the challenges can be worked around. It takes very little time, work, skill, and equipment to assemble a basic salad as compared with making spaghetti sauce from scratch. Some pieces of equipment are also super simple to use, like a crock pot (dump in the ingredients and ignore for 4-6 hours) or a Foreman grill (turn it on, drop in a slab of meat, wait 5 minutes). Communities can offer free classes in cooking, nutrition, shopping on a budget, etc. -- or better yet, a community kitchen for people who have little space of their own. A shared kitchen in an apartment building has the added advantage of encouraging residents to socialize with their neighbors while cooking and eating together. So there are options to fix poor dietary access, if people want to pursue them.