The bulk of these regulations came after Sinclair's "The Jungle", pointing out all of the problems at food processing plants (albeit in fictional form, but there was truth in fiction)

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Whether or not the US Government "should" care is kind of a settled case for over a century in my eyes.

Now that being said, "processed food" is such an incredibly vague category filled with obvious snake-oil salesmen and fake news that I loathe to give any of that much of my attention. But the government thinking about what foods are safe (or not) to eat? That's literally their job and has been for a long, long time.

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Now we can talk about where the regulations have failed, such as the Peking Duck exemption. (Peking Duck is delicious, and clearly safe. But by 1970s law / understanding of regulations, it was quite illegal). And its important to remember these cases and that regulation is far from perfect. But the overall concept of caring about what foods are safe and not safe is the point of FDA, USDA, and a whole slew of other agencies.

One issue with modern food production for consumers is that it can be difficult to determine whether a particular ingredient is safe or healthy. Information has public good qualities, and having each consumer make an individual determination about whether each ingredient is safe/healthy is costly. The obvious way to justify government intervention here is that it's more efficient for us to pool resources and empower a third-party to produce the information we need and ban/label products that are likely unsafe. The decisions won't always be right, but the approach might be much preferred to the alternatives.

Plus, conditional on socializing a lot of medical costs, it might make sense to put restrictions on behaviors that affect individual health.

The free market will take care of it, right?