Nye makes some good points. One of the reasons teachers tend to rail against the Common Core is because, depending on the specific school administration or school district, it can feel like we're pressured to teach to the test. (This is not something I've had to deal with myself, but I do know others who feel that way.) There can be little leeway in terms of assignment interpretation or the inclusion of activities or materials that could augment and strengthen instruction. Furthermore, teachers are often judged on how well their students perform on standardized tests, and are sometimes labelled as poor or weak if their students do badly; external and extraneous factors are not taken into consideration. The Common Core can be limiting and confusing, and doesn't always give teachers the ability to provide different or additional levels of explanation or instruction.
That said, I do think it's important that folks learn a broad curriculum that would include science, math, literature and writing, the arts, etc. I've seen homeschooling done well, and I've seen it done badly - folks who don't know their own limitations of what they're capable of teaching ("I'll teach physics AND chem AND advanced calculus AND graduate-level literature! Teaching writing isn't hard; I write all the time!"), or folks who don't think their kids should have to learn anything they don't want to (which can lead to multiple problems down the road, including a refusal to do something the individual doesn't want to do, poor basic writing skills, etc.). I can see why the Common Core is in place, but would like to see massive reforms take place.