You are the one who is truly confused between republicanism and democracy in the small "r" and the small "d" in comfort. To move direct decision making about government issues for elected representatives (republicanism) to the people themselves (democracy) is progressive left wing non-classical liberalism.
Let's work through this again. People deciding for themselves isn't democracy, it's freedom. Democracy is a way of making decisions as a group. What's really at issue here is whether decisions about curricula will be decided as a group, with everyone complying with the decision of the majority, or whether individual families decide for themselves.