My sense is that a lot of the reaction to the New Math was the same as the reaction to dictionaries that came out at the time and included words like "groovy" or said that it is not the crime of the century to write "which" when not preceded by a comma. That is, it was more of a political effect than pedagogical.
That said, there were a lot of people who thought on a lower or more local level, that math teaching that asked students to notice the Associative Law, or that asked students to think about sets, was not directed to immediate gain. That made it wrong in these folks's view, or at least puzzling. They had not learned those things and one thing about Math is that it is unchanging so ... .
I am excited by the Common Core, myself, but I hear a lot of rejection that seems to me to be just general opposition to doing it differently than when the speaker was in school. There was a lot of that in the 60's, for sure.
My evaluation of Common Core so far is that it's terrible. Students are forced to learn multiple ways of doing basic arithmetic, which is highly counterproductive and confusing if the child mastered the topic (and concept) with the first approach. It also lacks any aspect of what this essay stresses - helping children see the beauty and creativity of math. The emphasis is on learning processes by rote.
We'll see how things progress in the next grade, but I'm not optimistic. Private school is looking very attractive.