To key you in, Common Core isn't worth the political battle it's turned into. The Common Core standards are largely not terribly different (although one could mostly argue they are now improved) from the previous state standards. Furthermore, despite the pretensions from proponents and opponents, the Common Core standards are not going to substantively change (for better or for worse) the education system. When all is said and done, standards are just standards and they are just about the last place I would look to improve education. They are incredibly vague, they don't tell a teacher what to teach (or really how to teach) beyond generalities.
The recent populist outrage at Common Core is misplaced anger toward publishing houses which merely used the Common Core name to sell some eager districts. They also presume that the Common Core is responsible for national standardized examinations, even though this was the evolution from No Child Left Behind and Race to the Top. People that replace Common Core with their state standards aren't doing much better or worse than what existed beforehand.
Now, as to conservative by practice, I would argue that it is conservative. What was going on in the 1980s with education policy was two somewhat broadly contradictory ideas: 1) Increased emphasis on state and local control 2) Increased federalization in certain areas meant to address broad failings throughout the country. With the latter what the Fed Department of Education was doing was trying to ensure the best bang for the buck with taxpayer funding. They thought that for the money the public sends to these schools, we ought to ensure that they meet certain expectations in return. Then on top of that, they argued an even older traditional conservative message: the health of the Nation. By appealing to nationalism, conservatives were able to argue that the health of the State (especially against international competition) depended on an educated and productive citizenry. That's why the Federal Department of ED throughout the 80s released the A Nation at Risk Report, pushed standardized courses for graduation, demanded that schools better prepare certain populations for employment, and so forth.