Common Core appears to have become controversial primarily because the conservative media told us it is. Apparently they were hoping that the new standard would also find a way to further reduce teachers' salaries and career opportunities, and as it did not do that it needed to be destroyed at all cost.
It's more than that. The Conservative establishment has a vested interest in promoting ignorance -- creationism, global warming denial, abstinence-only sex education, poor nutrition, failed economic ideas, American exceptionalism, whitewashing of slavery and genocide, Christian dogma, the list is long. This is referred to as "local control": whatever whackjob ideas are predominant in your backward little community take precedence over what's being taught by all of those commie liberal college professors. The sad thing is that the teachers' unions are perfectly happy to jump into a disastrous alliance with these insane ideologues out of naked self-interest and terror that a few of them might finally be outed for, you know, doing a bad job. And children suffer.