So if the shared public school system is teaching something wacky (creationism or whatever), how should that be changed? If not parents or influential people, who would cause the change to happen?
School does not occupy all the time of a child. If a school teaches something parents disagree with, parents can debunk that when the kids are home. They can bring arguments and kids will eventually make up their own minds.
Homeschooling presents a single view, that of the parents. There is no debate.