I suspect that screaming down Rowling as a TERF is not winning hearts and minds, and at worst simply gives the right wing ammunition to depict the left as authoritarian and intimidatory.
This "debate" is between, on one side:
The entire print and television UK media (including supposedly neutral outlets like the BBC)
Numerous anti-trans lobby groups and "concerned parent" organizations
The current government
The courts
The far right and "culture warriors"
Religious organizations
The leadership of the independent body that is supposed to monitor human rights
A bunch of celebrities who decided to involve themselves
And on the other:
Young trans people on twitter
Right wing provocateurs pretending to be young trans people on twitter
A silent majority of queer people
A majority of academics and doctors specializing in this area (who are being deliberately excluded from the debate by the same people calling for more medical oversight)
Some LGBT charities
So yeah, I don't really care about winning hearts and minds. I don't see why anyone should. I don't see anyone talking about how the other side isn't winning hearts and minds when they post lists of trans social media personalities they think should be lynched or claim that trans women are all rapists because they symbolically rape women by appropriating female bodies, so why the hell should I or anyone care about whether we're winning hearts and minds? It's not like this "debate" is actually a real conversation.
Noone at this point is hating on JK Rowling because they think it's going to convince anyone. They're hating on JK Rowling because we need to keep reminding each other that this world is incredibly fucked up, that this isn't how it's supposed to be and one day it won't be like this any more.
In the meantime, it's not like most of the "hearts and minds" out there have ever actually read a single thing a trans person has actually said about JK Rowling, who by the way is a TERF. You don't get to rant about "rapid onset gender dysphoria" and "trans-identified individuals" and then not have anyone realise you're a TERF.