While I'm not fan of nanny states dictating what people can eat, vegetarians really are the bane of institutional food. The core of ancient peasant cooking was using the whole animal and that meant cooking things in broths, sauteing and frying animal fat, and using bits of dried/cured meat in soups/stews/beans/etc. Countless restaurants and cafeterias have switched to MSG-based bouillon and vegetable oils so that their dishes are vegetable-friendly. I admire Chipotle for standing its ground and continuing to cook many of their beans with pork bits/fat. This is truly sustainable cooking, utilizing regional products to their full extent. Some other favorite restaurants of mine, such as Momofuku, also explicitly do not serve vegetarian dishes because they buy and USE whole animals.
Contrast this with meat-free Monday, which encourages cafeterias to serve foods based on imported industrial monocultures (soy/corn/wheat) in the form of "textured vegetable protein" or "veggie burgers" and often utilizing industrial vegetable oils, additives, and flavorings. Or low-quality factory-farmed dairy, often with food coloring and sugar added.
I've eaten at many cafeterias and tend to think there is really no way to do institutional cafeteria food well. The goal is to feed a large diverse group of people as cheaply as possible. And the industry is dominated by a few large corporations that have deals with processed food companies.
And the cafeteria model in general may encourage over-eating and other problems. Since I was homeschooled until high school, I didn't encounter cafeterias until I was older. That's also when I started to gain weight. It's no wonder scientists use a "cafeteria" diet in experiments to induce obesity.
The solution probably lies with smaller schools with more local control and parental involvement. Unfortunately, the Federal government is increasingly dominating education, so this may be limited to private or homeschooling-hybrid* systems.
I think there is also a clash here between the politically-correct ideal of accommodating everyone and the fact that this accommodation often leads to inferior compromises.
*homeschooling hybrid is what I did for a lot of my youth. Some education is at home, but some classes/other stuff is done with other homeschoolers communally outside the home.