I work in a hospital kitchen/cafeteria. We function basically the same as a school cafeteria, just instead of students, we have patients.
I don't see anything wrong with giving a kid a sack lunch vs. hot, but I don't see how it's saving anybody any money. The food still has to be bought and delivered and the sandwiches and everything prepped by an employee. They throw away so much food in most cafeteria settings that a cold lunch may actually be an increase in cost as it requires prepping additional food to the hot food, some of which will likely be thrown away anyhow.
Your kids lunch is a bill just like any other. Parent's should pay up, but at the same time the kid shouldn't be punished if the parent didn't pay. You can't learn on an empty stomach and thats the whole point of school. My parent's frequently forgot to give me lunch money so I ran out many times. I was still allowed to eat, but I didn't get a free meal...the cost of the meal was just deducted from my account once there was money in it.
On another note, I went to an awesome junior high where the principal was a firm believer in a good lunch program. Everyday the students had two to three options for lunch: one or two hot food options or a cold sack lunch. The main hot food option was the same as any school lunch. The sack lunch was a sandwich (with many sandwich options to choose from), fruit, and cookies or chips. The second hot foodg option was usually pizza, hamburger, hot dog, or that sort of thing, and a bag of chips. This was not offered everyday. In this setting, the majority of the school bought lunch daily; rarely did kids bring their own from home. The sack lunches were quite popular, especially on days when the hot food was something weird, like fish tacos. I wish all school lunch programs were like that. Unfortunately I only went there for a year, then we moved. I started bringing my own lunches from home most of the time.