I'm not sure new math is to blame for anything. Traditional mathematics education had piss poor results. Eighth grade certificates and rote algorithmic algebra, geometry, and trig in the upper grades were not the fuel for high-level baccalaureate achievement. If you look at historical trends in numeracy and training in higher abstract skills like algebra and geometry, you see very little change in highest level of math training achieved (in the US, which has the same institutional gradgrindism as the UK). What New Math was part of, and what was the beneficial effect of the Sputnik shock, was the move toward opening up secondary ed to college level curricula - ie Advanced Placement.

I will totally agree that New Math is worse than useless when taught by idiots, but nobody learns anything from idiots. And number and set theory is very useful for hm setting the stage for analytical thinking later on, with algebra, trig, and geometry. Much of the base-number stuff, modular math, factoring, et cetera is to get the kids to do things with numbers that aren't strict computation. That gets the kids beyond thinking of numbers as grist in a problem.