Breakfasts aren’t provided as a form of babysitting. The prevalence of breakfast clubs in schools in low income areas should be a hint as to why they’re instituted. After school activities are more babysitting adjacent, but if parents can’t afford after school minding but really need to hold down a job, it seems like a great idea that also gets kids into sports or chess or whatever.
Why they choose to do it in school rather than providing parents with more money so they can afford to feed their kids is also a hint as to why they are instituted.
"if parents can’t afford after school minding but really need to hold down a job"
i.e. if you want to expand the workforce by providing child minding on the cheap through schools.