Let's suppose that children have a claim-right to high-quality education. This claim-right implies that someone else has a duty to provide a high-quality education for those children.
Generally, that duty falls upon the parents and/or upon the state. The state attempts to fulfill its duty by attempting to provide high-quality public education for all children. Parents attempt to fulfill their duties by (1) providing their children with access to high-quality, public education; (2) providing their children with access to high-quality, private education; or (3) providing their children with high-quality education at home.
Many people disagree with the priorities of these duties (for example, whether the primary duty is on the parents or on the state) and many people disagree with whether any particular example of education counts as "high-quality" (for example, whether a particular teacher is doing a good or bad job at teaching a particular child).
Some people don't believe that children have a claim-right to high-quality education.
My opinion is that a child's claim-right to high-quality education is only violated when neither the state nor the parents provide this high-quality education. States generally provide public education in order to relieve parents from having to fulfill this duty themselves (because of cost or incompetence), while some parents choose to provide this education themselves or privately because they believe that the state fails to fulfill the duty because the education is not "high-quality".
There are people who fall outside of all of these groups as well, and I don't intend to dismiss them. Most people, however, fall into these groups.
With this interpretation of the claim-right of children to high-quality education, it is irrelevant whether the child wants to be homeschooled.
Here's a question: do children have a power-right to waive their claim-right to high-quality education?