Tolstoy famously began his novel Anna Karenina with the declaration that all happy families are alike, and all unhappy families are unhappy in their own way. The “anti-gender” crusaders have yet to take this up as their motto, which is really a wasted opportunity on their part.
The idea of “gender” is disruptive to Russian conservative circles for the same reason progressives find it liberating: identity, behavior, and family structures are understood not to be “natural” or “innate,” but the product of social forces that obscure their own role in gender’s construction. Conservatives prefer to see all these phenomena as fundamental, yet still subject to perversion by unnatural influences. The result is itself quasi-constructionist: “proper” gender and sexuality are inherent and natural, while LGBT people and feminists have been re-constructed by malign forces.
The general conservative turn in Russian family and gender politics is not new. As feminist scholars have noted for decades, the alignment between feminism and socialism in the year immediately preceding and following the October Revolution narrowed the popular understanding of feminism in Russia. On the one hand, the “woman question” was considered solved by bringing women into the work force; on the other, nearly every difficult social problem involving the family could be blamed on “feminism," from the preponderance of single-mother-led households to juvenile delinquency. Moreover, the perennial complaint of educators and sociologists in the Brezhnev era that Soviet men were “infantilized” (deprived of their “natural” leadership role at home, marginalized within the family, and generally irresponsible) could be ascribed to the excesses of Soviet “feminism.”
In the 1990s, a hypermasculine business/criminal culture welcomed patriarchal norms, while growing demographic concerns (as well as a growing role for the Russian Orthodox Church) led to increased calls for larger families with a traditionally gendered division of labor. Still, it was the 1990s: the media and culture industry featured too many diverse voices for the anti-gender narrative to go mainstream.
A Real Boy
As in so many areas, the rise of Vladimir Putin would mark a turning point in the gender narrative, even if that turning point was clear only with the benefit of hindsight. And I am uncomfortable ascribing too much significance to the Russian president as an individual, since the American media’s myth of an all-powerful Putin micromanaging a vast country is both pernicious and lazy. Following on the work of Helena Goscilo and Valerie Sperling, I prefer to treat Putin as a phenomenon, a media/cultural creation that need not be identical to the actual man, nor reflect his personal intent. 
Viktor Pelevin notoriously suggested in his 1999 novel Babylon (Generation “П”) that Yeltsin and his entourage were all figments of the media imagination, created by political technologists with access to cutting-edge CGI. Putin, by contrast, is a construct meant to project manly authenticity; Pelevin’s novel went behind the curtain to turn the camera on the puppeteers, but the Putin phenomenon goes several steps further. A live-action simulacrum of his own self, Putin could join Pinocchio in a joyous chorus of “I’ve Got No Strings."
The first Putin/Medvedev decade, while definitely marked by an uptick in patriotic rhetoric and appeals to the national patrimony, nonetheless resembled the 1990s in the absence of any sort of national ideology championed by both state and media. Culturally conservative politicians were gaining influence regionally, but it was only with Putin’s return to the presidency, the crushing of the protest movement, and the troubles in Ukraine that “traditional values” would become the foundation of Russian public culture and policy.
 The exception here is the “born this way” mantra that has facilitated LGBT acceptance in the straight world. The insistence that sexual orientation not only is innate, but must be considered innate, has been the subject of critique not only from opponents of gay rights, but also from voices within the LGBT community. See Shamus Khan “Not Born This Way: Other Liberation Movements Have Rejected the Idea that Biology Is Destiny. So Why Should Gay Rights Depend on it?" https://aeon.co/essays/why-should-gay-rights-depend-on-being-born-this-way July 23, 2015 and Jane Ward. Not Gay. New York: NYU Press 2015.
 Mikhail Zygar makes this point very well in the preface to his recently translated book, All the Kremlin's Men ("Vsia kremlevskaia rat'"):