Hey, we're also planning on homeschooling our kids.
Have been doing lots of research in relation to it.
My spouse is an early-childhood-educator,
and I program artificial general intelligence.
You mentioned so far several "school-at-home" approaches,
such as correspondence schools and parent as teacher.
There is also "parent guided learning through non-textbook and life" style education,
such as woldorf, unit study, classical education and living books.
As well as "child-directed-learning" unschooling,
where the children's interests guide at least some of the material that is focused on.
We are likely going to do an ecclectic mixed approach "life learning with various degrees of structure".
Here is a good link with explanations about these various homeschool education styles:
Teaching Methods for homeschooling
Genius's are more likely to occur, if a child has many (mature) adult role models.
If they are surrounded by peers, that can stunt their learning, and behavioural development.
It's only in the last few hundred years where children have been institutionalized.
Living and learning near parents and seeing how the world really works,
is far more informative, for learning useful life skills.
When people, including children, see the usefulness in learning a skill,
they are much more likely to be interested, and integrate it for life.
Also another good thing to know about homeschooling is parenting-style.
Authoritative parenting is the best kind, for the most independent, happy and successful children.
The authoritative parenting style: A guide for the science-minded parent