It seems like the biggest correlation between overall health and processed food intake is that processed foods are calorie dense and lead to weight gain. If excess calories are coming from sugars and unhealthy fats, and one is consuming large amounts of bad additives, that will surely lead to bad outcomes.
What I would like to see is isocaloric studies, where individuals get plenty of fiber, micronutrients and a good macronutritional balance from "healthy" foods, but are also allowed some reasonable percentage of their daily intake to be processed.
Maybe it's the guilt of feeling like I'm a weaker person for being addicted to having desserts or chips on a regular basis, but I also feel like they are pleasures of modern life that are worth enjoying in moderation. I mainly strive to eat healthy for all my core meals, and allow myself ultra-processed snacks, while not gaining weight.