One way I find it useful to think about this is to ask, what might a good society for women look like, pre, say, 1920? Or pick some other year before the widespread availability of antibiotics and reliable birth control - where the female body we have is the body we have to accommodate.
Or to put it another way, is it only possible to have a good society for women if we can suppress, through technology, some of the major features of female biology? Personally, I don't think so.
One of the effects that the Pill in particular, and other more advanced birth control options, is the creation of the sense that sex is something we should all have access to, that within a relationship it's normative to have pretty continuous sexual access or the relationship is really done, there is even a pretty common idea among the sex-positive crowd that not being sexually active is a sign of a mentally and emotionally unhealthy person. Sex, for the vast majority of people now, is seen as some kind of leisure activity without great consequence.
That is a really different viewpoint than has existed before for most people, where sexual desire was seen as something that had a place and purpose, but had rather significant consequences, and therefore had to be managed. And it was recognized that managing it wasn't easy, because it was a powerful drive, but that it was necessary.
I get the sense from a lot of people now that the idea of really having to manage sexual desire is kind of an imposition. (By nature, I guess?)I don't think that's disconnected from other ways in which people have come to think about the restraints of nature as an unjust imposition.
Or to put it another way, if being really free, as a woman, requires the technological suppression of female biology, what does that say about woman as an essentially biological category, or about suppressing other elements of human material reality in pursuit of freedom? Which is why I think Harrington is drawing a link between the Pill and transhumanism.