I am not so sure that argument is really clear. After all, it's not the political right where you see parents and individuals falling prey to this in large numbers.
I also think that GC feminists really over-estimate the extent to which people have a hard belief in gender being something that could or should be abolished,while at the same time tending to over-state how rigid people, especially conservatives, think that gender roles should be or are.
I think the vast majority of people believe what GC feminists seem to think is a conservative position: that on average, there are some temperamental and behavioral differences between male and female humans (just like most other mammals,) which are biologically based, but also that there is quite a lot of overlap.
That differences like reproductive role and physical differences also tend to lead somewhat to different ways of behaving and thinking, because we are embodied beings.
And that taken together, these temperamental differences, and the consequences of things like pregnancy and motherhood, do lead to an overall somewhat different set of social processes that are common around the life of men and women. That doesn't mean there aren't lots of variation, or that people are compelled to do anything in particular, or that anyone should be barred from any particular career choices or whatever. It just means that there won't ever be a society that isn't in some way shaped by the differences that come out of reproductive role, and that is ok, and can even be in some ways supportive.
Matt Walsh might be somewhat on the harder end of this spectrum, though I don't think spectacularly so, and in many ways I suspect his view is closer to that of the vast majority of moderate people than the GC view is.
These people, and I would include Matt Walsh because he is really taking what is a perspective that comes out of the Catholic view of the body-soul unity, don't see that these differences mean you can "be born in the wrong body." Because that's backwards - the differences, although they become culturally reflected in various ways, are essentially the differences that tend to emerge from the kinds of bodies people actually have. It's not some kind of Platonic abstraction. A woman who likes car mechanics or Rush is not a man just because some of her interests are more typically shared by men, or even a woman who really hates being around children. It's about having a female body, which may mean she is less likely to be a big Rush fan, but would certainly mean she is necessarily subject to different material realities than a man particularly around her reproductive role and all the things that touch on that. And that's significant, and for many many women will shape their lives and decisions differently than if they were male.
This idea that conservatives and especially religious conservatives think gendered social roles are somehow separately set down from embodied causes and are really arbitrary is really mistaken. That kind of dualism is from a quite different source, I think, and not one particularly exclusive to the political right or conservative thinking - it seems to belong at least as naturally to the left, and the idea that the sexes can completely transcend embodies differences is a significant example of it. It's part of why people like Walsh tend to see feminism as having led people down a garden path.