Home schooling already has a long tradition in this country. It has become a lot more widespread in recent years. Back when we had Glades Herp in the '90s, we had some home school groups come through for tours, so got to talk to some of the parents. They were so much more involved with their kids and in education. It was so refreshing to talk to them compared to many parents of public school kids.
Charity is great! But it wouldn't really take much, if any, charity for a parent to home school their child. It takes time and effort. If they are really poor, the public libraries have computers available for them. (maybe public libraries will remain public, or maybe they will be replaced by community funded libraries in much the same way that HOAs fund community playgrounds. With more and more books and resources available digitally, a small, community library may not cost nearly as much as traditional libraries do).
Many states already have online, free, public school classes available. Maybe at the beginning, these could remain publicly funded, and eventually figure a way for the private sector to take them over. Same with the libraries. I sure don't advocate just dumping every government service all at once. First, get rid of the services that are the least needed and waste the most money. Once we figure out the most efficient way to privately deliver those services, we can tackle other issues. Some of those may need to remain in the public sector if it turns out that is the most efficient way to accomplish those goals, but most could go private eventually.
BTW, home school parents told me that their kids do have to accomplish certain goals each year, and are tested to be sure they are on target.
Home schooling and private schools are not novel ideas, and their students often excel in college compared to public school grads. Only the scope of replacing ALL public schools with them would be novel, and would have to be worked towards one step at a time.