From a homeschool perspective...
No, I don't think it should replace public education. Being a homeschooling parent requires certain things that not all parents are equipped for or willing to do. You have to be able to invest time in your kids. You have to care about their education more than just wanting them to get good grades and going out for pizza at report card time. You have to consider their strengths and needs, their interests, and create a learning environment and experience unique to your child.
Some parents are just too lazy or busy or caught up in their own lives to put that effort in. Those kids need public education so that they have a chance to succeed in life.
Public education definitely has a place in our society. I just wish that it wasn't considered the standard. The truth is, an education tailored to fit your particular child's needs and altered as needed to continue to do so is much more conducive to learning than the public education system's rigid, test-score driven, trying to shove everyone into the same mold, factory-like style.