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.
The public school that my daughters attended last year did not have the subjects that I listed, or they only offered a limited version of them. They had history for a month in May after the state testing. They had only a few days of science. The art was a volunteer program run by parents and it was about 6 times during the year (but it was a good program (Art Vistas), and I think better than the art they are getting in private school).
Our district closed the school that my kids attended, and moved the kids to another school. The first school was a National Blue Ribbon and it was okay, but not as good as the private. They at least had more history, science, music, and PE than the school they attended last year.
The new school had lots of ESL students, and they were more interested in helping them instead of the special needs kids or the gifted kids. (One of my daughters is special needs, and the other one is gifted.)
Plus, both of my daughters actually do well on the state testing which means that they both got ignored because they tested well. They never got rewarded for doing well. Only kids who improved their test scores or improved their behavior got rewarded. Kids who already tested well, and had good behavior did not do well.