Flaming-hot take, fully open to debate this. I'm not here to point fingers or assign blame per se, but to shed light on a massive strategic error on the part of feminists that has led to terrible consequences.
Third-wave feminism has two important beliefs:
Rejection of essentialism (the idea that women and men are fundamentally biologically different)
Masculinity is toxic/problematic/immoral/otherwise stigmatized
These two premises, taken together, lead inevitably to trans ideology.
Once essentialism is rejected, there are no moral grounds to differentiate between males and females. Without a fundamental biological basis to sex, how can you say that male wearing a dress is not a "woman"?
Where rejecting essentialism provides the opportunity, stigmatizing masculinity provides the reason. Take a batch of awkward males, tell them their masculinity is problematic and men are toxic, set up women as the admirable opposite of that, and remove all barriers to males being seen as "women". The effect is as inevitable as entropy - particles will freely move from a high-pressure environment to a lower-pressure one. The effect is even more pronounced when many actual women, in the name of progressivism, encourage them to do so.
The strongest evidence of this comes from the political beliefs of trans-identified-males themselves. They are almost universally left-wing progressive activists, to the point that it seeps into their daily lives. This is exactly what you'd expect to see if radical progressivism leads to becoming trans, or if the two movements are connected in some way. TiMs and third-wave feminists agree on almost 100% of issues.
This relationship is reciprocal as well. While many TiMs support third-wave feminism, many third-wave feminists also support trans ideology. I'm sure many here have encountered actual women who are outspoken feminists, but later reveal themselves to be vehemently pro-trans as well. And it's well-known that progressive circles promote trans ideology: consider this post shared on anarchy and lgbt subreddits, or this article equating biological sex with "white supremacy".
What are the solutions to this? One would be to double-down on biological essentialism. Here is one great example of that, while here is another. Another solution would be to de-stigmatize masculinity, and tell young men that it's okay to be a man. I'm surprised more gender-critical feminists haven't gravitated towards figures like Camille Paglia or Jordan Peterson, both of whom are critical of trans ideology and recognize the need of young men to attain a sense of masculine self-worth and confidence (lest they be taken in by far more subversive ideologies).
TL;DR: As long as biological essentialism is rejected and masculinity is stigmatized, the transgender movement will continue to grow.