In the first place, the impetus to "reform" math education was motivated by politics, not by any serious observed deficit. By coincidence, there was a "new" style in higher mathematics, reflecting the previous 50-60 years assimilation of set theory and rewriting of many things in terms of set theory. But until the "sputnik scare" no one had incentive to pretend to incorporate set theory or not. The people who "decided" to promote that "New Math" were mostly not mathematicians of any sort whatseover, but, rather, semi-politicized people who needed to be able to "show that they were doing something". After all, set theory was not new, was not what had made any difference in WWII or subsequently, nor was it what made the Moscow school of mathematics what it was.
Many parents objected not on scientific grounds, although it was veiled as such, but because the "new math" was alien to them, and kids who were learning whatever "new math" purported to be were not learning "traditional math". Even though "traditional math" included (I was there...) an enormous amount of repetitive drill, arguably to the point of senselessness, it was orthodox and familiar. Parents could no longer help their kids with their homework...
More substantive than parents' discomfort (though that might have been the dominant political determiner) essentially nothing had changed at the college and university level (or even high school), and kids that knew something about set theory couldn't do the basic algorithmic math for basic chemistry, physics, nor the standard high school math curriculum (whatever the flaws of the latter). Certainly "the new math" did not warm people up to the traditional trigonometry, calculus, etc.
A more insidious problem was that few of the elementary-ed teachers (nor middle-school, nor high-school) had prior preparation in such stuff, ...
In fact, of course, if there were room in the curriculum for it, and if kids were ready for it developmentally, both some sort of "meaning" and "algorithms" could be taught.
But there's neither room in the curriculum, nor (in my observation) are kids ready for more conceptual things at that point. Perhaps it is harder to teach concepts than algorithms, also.
I was in school then, and my parents both taught high-school math, so I heard a lot about this. It is important to note that, for better or for worse, "school boards" (whose qualifications are mostly political) decide textbooks. The "new" books my parents brought home were more interesting to me than doing yet-more elementary arithmetic, which I could already do, but it was also clear that most of the other fifth-graders wouldn't have been in the right state for such stuff, since they were still having trouble "following instructions" about very concrete activities. (I do not have any citations on "development"...)
But, so far as I recall, it's not that there was scientific-grounds objection to set theory itself, but that it displaced indispensable things, ... and was new and scary to non-mathematicians. Also, some of the semi-politicized proponents were blitheringly incompetent math-wise, which made them amusing targets for actual scientists... even though that was not the same thing as a criticism of set theory, etc.
So, with just a nudge from practical issues, the opportunity for political action and argument both created and killed off "new math". (More recently, an attempt to modernize calculus and related material has re-generated "math wars", again with similar extra-mathematical and extra-educational factors dominating the action, ...)