It's also possible that their defenses sound like that to the people making the "not literally illegal to say them" crowd. As a stereotypical example, grab a random Christian, they say they believe gay marriage is wrong on Biblical grounds, this gets interpreted as "I want to hate gay people".
If we look at racist hate speech, people literally lecture that people who happen to look like me are "born into not being human", say the lives of people like me don't matter. We are told we should not speak and yet that our silence is violence. Trainings for "diversity" across a wide range of institutions have started segregating along racial lines, and the list goes on. During the Grievance Studies hoax, for example, one of their papers was sent back because it was too sympathetic and not sociopathic enough. Said paper advocated making white students sit on the floor and putting them in chains as a learning experience.
Yet open racism is fine, because the target ethnic group and sexual orientation are acceptable to the activist class.
Likewise, people who get kicked out for "transphobia" will often not be people who hate transpeople in any proper sense, but have policy disagreements where there are no win-win solutions. Yet anything but being completely on board with the activist class's 100% no limits pro-trans policy agenda is "phobia". The claim holds up as well as wet paper.
How about gender affirming treatment for kids who cannot physically know what sexuality properly is because they haven't been through puberty? Someone's a bit off their gender role? Tomboy, nah, they're a boy, time to stuff them full of hormones. Concerns that this just might be insane are again, labeled hate.
Said cultists insist on mantras of trans(wo-)men are (wo-)men in an explicit denial of biology that the best of our medicine can't alter: We just craft a facade that eases dysphoria to care for our fellow human beings. As Buck Angel put it: "I use testosterone to masculinize myself so I feel more like me." He's routinely called a transphobe.
This is basically the biology equivalent of flat eartherism or young earth creationism, but not subscribing to the creed is stamped hate speech. The mantra wouldn't even be necessary if its claim was actually true. Hell, it's arguably even worse than flat eartherism, since a human can't at a glance confirm that the Earth is indeed round, but TWAW/TMAM asks us to baldly deny the input of our eyes in everyday life.
It's not one side that has gone insane: Good chunks of both are open lunatics. Q at least is thankfully fringe, but even they've managed to get representatives elected.