I have a friend that home schooled her threee children. She was not equipped to do so. She just didn't have the fortitude to make the kids do work. As a result, her kids are not productive. Roaming from dead end job to dead end job.
Two of the three still live at home at 30 and 28 years old! She simply cannot let go of those apron strings.
There are plenty of institutionalized children with parents like the one you described who aren't doing well, either. Ask any government teacher who complains about the lack of parental support.
It has been my observation ( anecdotal) that academically successful children, whether institutionally or home educated, have parents that are equally committed, equally engaged, and are spending the **same** amount of time providing homework support.
My conclusion: It could be that institutional schools are not teaching much. It could be that it the parents and children in both cases, institutionally schooled or home educated, that are doing the real work **in the home**. The institutional school may merely be sending home a very expensive curriculum for the child and parent to follow in the home.