>>Right-wing and centre-right men (and women) will listen to Matt Walsh, whereas they wouldn't (at least until recently) listen to left wing women, like JKR or any feminist trying to bring this subject to a wider audience.
That's not my experience. I consider myself Right Wing and I move in Right Wing circles. JK and others were frequently quoted on the subject. I was first introduced to Magdalene Burns by friends who I would certainly describe as Right Wing to very Right Wing. And that was years ago.
In so far as there is difference there can be a bit of attitude of 'feminists brought this on themselves by championing Identity Politics and depowering men from dealing with trans women themselves" but that hasn't stopped gender critical women being quoted and referenced in Right Wing circles for as long as I've been in them.
>>So there is an element of a pincer movement in trying to get the left to listen to certain people with a profile, and getting the right to listen to the same messages but from someone else.
Again, I'm sorry but not my experience at all and I have moved in Left and Right circles a lot over the course of my life. I have always found the Conservative Right to have a better understanding of the Progressive Left than the other way around. In fact, there was an interesting study to this effect where Left and Right wing people were asked what the other would think about various matters and the Right usually were much more accurate about the other party than the Left were.