I haven't read Lindsay's book yet, but I think there are some problems with the argument that feminism and gender identity theory are "rooted" in Marxism. You can easily cite early feminist texts like Mary Wollstonecraft's A Vindication of the Rights of Women (1792), written and published before Karl Marx was even born. Just from the title, that's very obviously rooted in the language of human rights that comes out of The Enlightenment.

And a fairly clearly-articulated idea of gender identity was floating around among the early sexologists in the late 19th C. like Krafft-Ebbing who talked about the idea of inversion, "a masculine soul in a female body" or vice-versa. Looking back, it was always obviously an attempt to conserve heterosexuality and stereotypical gender roles, as it often still is. In other words, a man attracted to other men must really be a woman. That's where the idea of a man trapped in a woman's body comes from.

I did research in 19th C. medical ideas for my dissertation, and the latter part of the century is rife with crackpots and crackpot ideas about gender and human psychology.

Lindsay is probably correct that these ideas get packaged with Frankfurt School Marxism in the period since the 1960s, but they have distinctly older and different roots.