We had cafeterias but they were by and large for treats not for day to day lunch.
I don't understand why taking their own lunch is not an option if the food served is so abysmal?
Just because you don't agree with how a parent decides to feed their child, doesn't give you the right to arbitrarily take away their choice.
It's not a simple decision - here in the UK I would have my kids eat school meals most days (perhaps miss the meal they anticipate to be least appetising and instead make their own). Instead they have packed lunch [aka bagged lunch] nearly every day because we can't afford the school dinner.
I'm afraid it is really necessary to provide a decent school meal for many children in order to ensure they have some chance at a moderately healthy meal once a day.
On the other hand, if it's chore to take the time to prepare and cook their meals, and if school was presenting an affordable alternative, why wouldn't use it?
No-one is forcing the kids to eat at school, no-one is taking away the parent's choice.
The top four sources of calories in the average American child’s diet are grain-based desserts, pizza, soda and sports drinks, and bread. One-third eat fast food every single day. More than 90 percent don’t eat enough vegetables. And each year, our children are bombarded by around $2 billion in child-directed food and beverage advertising, much of which promotes the least healthy products.
If the kids bring their own lunches, those lunches will be crap. I can supply some anecdata on this. My mom taught high school and observed what the kids brought to lunch. I saw it when I did things like chaperone for field trips when my kids were in grade school. One kid's lunch was a sack of cookies. And if you want to be a snob for a while (disclaimer: I'm a bit of a snob), look at what people put in their shopping carts at the supermarket.
And it's quite noticeable that affluent people buy seemingly healthier food, and are much less obese, likewise for the lunches that their kids bring to school and the physical condition of those kids.
And it carries over into adulthood: A large fraction of my colleagues eat restaurant food for lunch, at least a couple times a week. The vast bulk of restaurant food is a pile of sugar, fat, and salt.
In my view, the US is not a fancy food country, and what works for us might not be fancy food, but teaching people how to prepare healthy food with modest ingredients. For instance I suspect that a sandwich and an apple would be a vast improvement over what most kids eat for lunch. That's all I ever got in the lunch that I brought from home as a kid. When I got old enough, it was my job to make my own sandwich.
As you indicate, this depends a lot on class.
Anecdotally, I always begged for the pack'ed Lunchables at the grocery, instead, I had make my sandwich with mustard, lettuce, meat, and cheese.
I often wonder if we (parents or schools) taught kids how to make their own lunches again if it would drastically help the health of most families, but especially impoverished families.