I think that part of the reason how the reader, particularly a female one, can find resonance in Gilman's work is that it strikes at the basic fear that many women experience. The issue of post partum depression, something that Gilman herself experienced, the idea of a social order that refuses to acknowledge that something is indeed "wrong," and the paternal silencing of women's voices strike fear in any reader. These conditions are particularly relevant to a female reader, who finds that her situation in comparison to the unnamed narrator might not be entirely different. The image of the woman isolated from her own child whom she delivered, contained in a room with no contact with the outsider world, as well as denied means of self expression are particularly scary conditions for any woman to experience.
Gilman is very careful in her depiction. She is not suggesting that the images she presents are how women are or what women should be. Rather, she is focusing on the condition of a voice being silenced. The narrator's voice is being denied by the doctors, by her husband, and by a social order that continually says "everything is fine" and "nothing is wrong." Gilman's depiction is one in which an experience is presented and it seeks to broaden out to the reader's acceptance. It is very likely that a female reader is going to be receptive to this depiction, only because it is part of a historical narrative about what it means to be a woman in America. It makes sense that the story was "rediscovered" in the 1970s, when there was a social groundswell of support for feminism and broadening solidarity amongst women. It is in this spirit that the work speaks to images of women in society. The work is not suggesting that women are what is presented, but rather presenting a narrative in which women's voices are silenced. The reader comes to understand that this is a part of a social and psychological narrative that has to be understood if it to be avoided from being repeated.
I am not sure that it is "coopting" women readers as much as it seeking to broaden experience with them. Millions of women experienced the denial of voice regarding post partum depression. Gilman is, in a sense, "preaching" to ensure that such a condition is understood and validated. It is in this experience of broadening voice and forming solidarity that the female reader could be persuaded to embrace the ideas presented.