One big advantage of homeschooling is the ability of a child to really focus on an activity and expand their expertise. This can be quite attractive to any number of Ivy league schools.
A very good friend of my daughter discovered she LOVED languages. In her Sophomore year, she transitioned to home schooling to allow her to really concentrate on languages not offered at any public school. She became proficient in about 4 and gained a working knowledge in several others.
Full ride to Yale.
and at Yale she got liberal indoctrination probably