There's no contradiction, you are misunderstanding my statement. Older children need to go to school to learn stuff like mathematics, history, sciences, languages, and so on. Hanging around and taking care of younger kids all day would be a hindrance to their development.
It's too much to assume that an average child can productively solve problems that are related to raising children with various family backgrounds. How to deal with shy or angry children, conflicts with the family members and so on. While I don't know how it's done everywhere in the world, contrast that to kindergarten teachers that have academic education with focus on teaching (including methods, subjects, psychology, and so on).
If you claim that education hasn't advanced in 100 years tremendously, you have absolutely no idea what you are talking about. For instance, think about the education of children with mental or learning disabilities. It has advanced a lot in my lifetime, too.