Terribly sorry! It wasn't a lack of bother to read the whole article...I just somehow got distracted and didn't scroll to the bottom (but believed that I had). My mistake. You did provide references.
The references do, however, only actually do what I say (if you look carefully): debunk correlations that were never causal, debunk strong dichotomies by finding counterexamples without showing whether there is, nonetheless, a trend, and the like.
The reason I say neuroideology is foolish is because ideology kind of by definition isn't based in conclusive evidence. Furthermore, drawing links between ideology and neuroscience is fraught with potential for rationalization and bias because the degree to which we understand the mechanistic basis of the brain is inadequate to make any such links particularly convincing.
For instance, separated twin studies show that there is apparently a very high degree of genetic and/or epigenetic specification for cognitive function of all sorts--there are strong correlations in IQ, in big five personality traits, in sexual preference, and so on. And there are striking sex-specific expression differences in all tissues (including the brain).
So we have plenty of evidence that mechanistically there could be almost any degree of sex-specific differentiation in the brain. We also have plenty of evidence that there is a huge impact of genetics and/or epigenetics. But we also have plenty of evidence that the brain is highly plastic (e.g. the London cabbie study), and that society can shape how people use their brains. So it's also consistent with development and experience being a major factor.
The Woodhill and Samuels paper does not endose the "brain as sponge" view. For instance, "Although still the topic of much debate, small to moderate gender differences show stable and substantial differences in more than a few areas. Studies clearly demonstrate that genetic and hormonal components of sex can affect the structure and function of brain cells in utero and throughout life".
Professor Rippon's book again gives many examples of overzealous interpretations, but provides almost no information about what the sources of variability are (just raises the possibility that what was thought to be an innate sex-based difference is instead a socialized difference...but it does not show that the magnitude of the effects can be explained by socialization anywhere that I found, at least).
The Medium article by S. Leguichard correctly criticizes sloppy research but without making any substantive case for a different point of view. Concluding "biased, therefore wrong" is a variant of the fallacy fallacy. The correct conclusion is that we don't know what we thought we knew, not it is the opposite of what we thought.
So I think your central claim of subtle cues influencing a sponge-like brain ranges somewhere between implausible to known-to-be-wrong. Granted, this is not your fault alone (Rippon does it too), and the faults in the gender/sexed-brain camp are at least equally numerous. But errors by an opposing camp notwithstanding, your view is not well-supported by evidence.