Don’t think they wont go so far as to outlaw home schooling. This is no longer the United States of America.
Yes, they will and there have been instances where they have already tried to make home schooling very difficult. They will sick social services on home schoolers, they will audit what you are teaching, and if they can figure out how, they will take your children from you "for the greater good." They are all in and will do absolutely anything in their drive for a New World Order IMO.