It appears that the New Math failed to catch on as a math reform because:
1) Starting with formal system and building from axioms on up is appealing to mathematicians but doesn't match children's psychology, which requires concretions rather than abstractions in the early years.
2) There wasn't a good enough plan to get parents on board with the reform. Mathematics was highly regarded yet poorly understood in U.S. culture, leading to pushback from parents (and teachers) when they saw a different mathematics curricula then what they had ("You can't change math! Math is Math!").
Math reforms will need to avoid these two snares. Perhaps a data science class at middle school and up would work well since it would avoid concerns about radical changes to the curricula.
One approach I'm considering is doing a history-based approach to mathematics. I don't have anything shippable yet, but explaining the history of the integral calculus has helped my students understand the difference between Leibniz's and Newton's notations, as well as provide motivation for why we're studying limits.