Leaving education to state and local governments is the quickest way to go back to earth being roughly 7000 years old being taught in school (as just one example of what is likely to happen.) Everything from history to science to economics would end up impacted and subject to whatever the local prevalent beliefs are. What is taught in California to Utah to Georgia (again examples) would get that much further apart.
It is reasonable to assume limited involvement in education by the federal government and more state or even local involvement in education might enable state and local politicians to bias the curriculum in a politically expedient manner. For anyone with an interest in science, the fear obviously concerns religious dogma, as you pointed out. It is also something which does have some precedent. I would like to add that some conservatives might express similar concerns, though more often with centralized planning than with state and local planning.
In the 1970s, the introduction of mandatory sexual education was heavily criticized by conservatives since, in practice, the curriculum was not directed towards the biology and psychology of sexuality, as much as it was used to instigate "healthy attitudes" toward sex. When you tell children or teenagers that some sexual practices are normal, acceptable and perfectly healthy, you are passing a normative judgment which might be fine if the intention of discussing ethics was made clear. As much as I personally find socially conservative attitudes regarding homosexuality appalling, for example, I find throwing ethical comments about the moral status of homosexuality in the middle of a discussion on biology and psychology to be extremely distasteful, if not outright dishonest.
Today, things go much beyond comments regarding the morality of being homosexual or of pursuing promiscuous encounters and are much closer in spirit to the outright denial of empirical evidence found in fundamentalist circles. Gender identity, gender expression, sexual orientation, and biological sex are extremely correlated variables empirically. For example, the vast majority of men present themselves in what is culturally associated with men, declare themselves to be men and are attracted to women. The same holds for women. In every case, you have pair-wise correlations in the 80s and 90s, which is irreconcilable with the social constructivist doctrine at the core of the identitarian left. It doesn't mean you have to attack people for being an exception to any or all of the above at all. There is also ample research in psychology regarding the differences in personality and other traits across men and women. The results suggest we are more similar than different, though we differ in systematic ways (i.e., the same ways across all surveys and across cultures). The average woman is more neurotic (she responds more strongly to negative emotions), the average man is more aggressive, the average man is more interested in things than in people, the average woman is more interested in people than in things, the average man is likelier to pursue multiple casual sexual encounters, the average man responds more strongly to visual sexual stimuli, the average woman responds to more abstract things like social status, confidence, etc.
If you are curious, Jonathan Haidt and Steven Pinker have commented in various books on these questions. Both call themselves liberal. Haidt also gives clues in his books that he supports Democrats more than Republicans. In fact, Pinker has an entire book ("The Blank Slate") where he slams the more radical people on the left for holding on to scientifically incongruent beliefs regarding human psychology. You can also find some if you look through the work of Jordan Peterson. Part of his expertise concerns research on personality and he has several papers on the subject. I didn't check for a while, but he used to direct a Ph.D. student regarding links between political views and personality, another very interesting subject.
The bottom line is, some conservatives might legitimately fear that leaving education in control of the federal government will put power in the hands of bureaucrats in federal agencies. These people are unaccountable, they also tend to be chosen for their zeal and they all lean in the same direction. It is not a problem without precedent either. I cannot recall if it is in the US or Canada, but wonders like the gender unicorn meant for children speak volumes about the fact stupidity is not monopolized by anyone.
