So much to say about this article, and I'm not a homeschooler.
First, I have a special needs daughter. Last year, she had a huge evaluation by a neuropsychologist. The neuropsychologist told me that homeschool was a great option. My daughter has speech and social problems, so I am very concerned about those areas. The neuropsychologist then asked if we went to church, did any after school activities. She said that there are plenty of socialization activities after school.
Second, I have a degree in engineering. I have more math and science than most elementary education majors, and probably more than most middle school teachers. There's only a few subjects that I would not feel comfortable teaching (foreign langauges, physics).
Third, if the public schools were doing their jobs, then people wouldn't be pulling kids out of them. Lots of public school do not have science, music, art, PE, or history. If you want to have those things, they need to be in private or homeschooled.
The most demanding job in education today is in special education...so there may not be a public school to help your daughter in your community. However, I have yet to find any public school not offering the subjects you list. I do, however, support your notion of private schools as I believe them much better at teaching than homeschooling.