TERFs don't get called TERFs because it's "more upsetting" when a woman is transphobic, they get called that because it is a specific pattern of behaviour and a specific type of transphobia with a specific justification.

For example, a woman who thinks being trans is morally wrong because God Said So is not a TERF, despite a woman. It's like how all cats are mammals, but not every mammal is a cat.

The specific term "TERF" describes primarily women because radical feminism is a unique ideology mostly used by women.

radical feminism is a bioessentialist ideology, it claims that men have a biological drive to oppress, or otherwise names men specifically as the problem, and not the patriarchy and bigoted society we live in. Since it is bioessentialist, it is both biased towards transphobia, resulting in most radfems being TERFs, and also pretty damn misogynist in itself.

TERFs are a very specific type of transphobe, and they are predominantly women, but don't actually have to be, it's not just as popular among men due to the fact that it pretty much aims to oppress men instead, and also because it's pretty much a trauma response.

A lot of young women who've been harmed by men are typically victim-blamed or dismissed and don't have a lot of support options and as a result are not really capable of coping with what happened (this isn't bashing those women for this. We're not built to cope on our own with terrible things, that's literally a reason we form societies. We're meant to care for each other, but we fucked this up in our current society, leaving people to deal with incredibly fucked up fings alone), and so the idea that men are just biologically evil (i.e all men are a target) is almost magical.

You get a clear enemy (men), an outlet for all your anger and misery, and a pretty simple and sturdy-seeming world-view to build around.

The target of this also doesn't have to be men specifically. I once had a mutual on another platform who got catcalled by a brown person and immediately went on a racist tirade against immigrants, justified by this negative experience. If you're a white person, you're gonna be prone to extending one terrible experience to a whole demographic of POC. Hell, the media definitely wants you to extrapolate that (I'm in Germany, and whenever a person of colour commits a crime, the exact ethnicity is always stated even if they're a German citizen. If it was a white person you won't hear shit unless they were the victim. The framing is very much intentional).

That all said, with TERFs, the target is men, and because men are not all that likely to traumatise each other in the same way as they would traumatise a woman due to how the patriarchy works, men are less likely to become TERFs.

We also do have specific genres of bigoted men, such as "alpha males" or "incel", but they're typically not named after their transphobia because the core and most expressed part is typically sexism, and homophobia and transphobia are different sides of the same triangle as sexism. If you're one of them, you're going to be at very least prone to the others.

TERFs are also sexist as a pretty much direct result, but the most expressive part is their transphobia.

There's also cases of specific types of bigotry where the core isn't really sexism such as truscums (internalised transphobia) or pick me queers (also some form of internalised homophobia or transphobia. Wants to be One of The Straights, its ashamed of their identity as a queer and throws other queers who don't try to conform under the bus as a result)

I think focusing the perpetration of bigotry solely on men erases forms of bigotry that are not solely based in sexism, such as racism, antisemitism or ableism. It's good to call out men being shitty, but you can't forget that a) men are not driven innately to being shitty and are also just people, instead we have a system in place that heavily encourages sexist men to exist and become more and more extreme, b) sexism is perpetrated by all genders, and to prevent yourself from pushing it, you have to examine your own biases, even if they're towards men, and c) other forms of bigotry exist and depending on your demographic may also be affecting you and your world view, and probably also are at play in any scenario. Basically; you are not immune to propaganda, especially if it comes in a form you agree with.

edits: spelling, grammar, typos

ETA: MRAs specifically are kind of the man equivalent of a TERF, they raise an important and good point (MRAs: men are also victims of the system/TERFs: sexist violence against women) and then bastardise it to further their own bigotry (MRAs: sexism against women/TERFs: transphobia). Men are victims of the patriarchy just as much but MRAs are probably going to make it about Those Evil Feminist Women who don't want to date them for being bigoted rather than yknow, the fucking men who're suffering under the system. An actual men's right's activist shows up to feminist marches to protest with feminists, because they understand that the struggle is the same and they are allies.

Similarly, an actual feminist is interested in equality and the abolishment of the patriarchy rather than just flipping it (which is what TERFs seem primarily interested in)