In Minnesota, the state pays for all K-12 students to get breakfast and lunch, no questions asked. It's cost ~$250 million a year, and it's worth every penny.
Figuring out what kids will eat has always been a challenge. Some districts are better at it than others.
The schools I grew up in and spend time around never overtly discouraged cold lunch, but sure do and did make it hard to get it if you did. No access to lockers, no cold storage, and 30 minutes desk-to-desk to eat. Also, there was only 1 choice for hot lunch every day. That was it or you didn't eat.