How discredited is this idea, precisely? What data do we actually have, particularly with relation to a theoretical law and situation that doesn't yet exist (as per Rowling's post)? But see below.
So, I don't know how familiar you are with the whole situation in the UK, but some necessary context.
Back in 2016, there was a review of the laws related to trans people in the UK. One of the findings was:
While we recognise the importance of the Gender Recognition Act as pioneering legislation when it was passed, it is clear that the Act is now dated. The medicalised approach regarding mental-health diagnosis pathologises trans identities; as such, it runs contrary to the dignity and personal autonomy of applicants.
Based on this report, the government announced a consultation on the gender recognition act in 2017.
The result has been an enormous cultural and media backlash against trans people, which has essentially continued to this day and which has gone far, far beyond any specific changes to the gender recognition act and become a debate about trans inclusion more generally. Much of the language used, even in print media, has been fucking horrendous, evoking the moral panic around the AIDs crisis of the 80s and 90s. Children are being brainwashed by the "trans cult", or "sacrificed" to appease "militant" trans activists. The medical and scientific establishment has been "coopted" by "demented trans ideology" which denies basic biological truths about human beings. Women are under threat. Gay people are under threat. Childen are under threat. Society is under threat.
If you read Rowling's essay as an isolated document, it probably sounds quite reasonable. Read in this context, it's simply a repetition of the same talking points we have heard over and over again. It's also very clear that she is borrowing language from trans-exclusionary feminism. She describes people as "trans identified", for example, which is a term TERFs use to describe all trans people because they don't believe that trans people actually exist. Above all, there is the same recurring sense of threat that must be managed.
Because that is what trans people are to 'gender critical' liberal feminists like Rowling. They can like and even care about us as individuals, but we are not and will never be normal people to them. They reserve the right to decide whether to think of us as the sex we claim to be. They expect their feelings of being threatened to outweigh our needs regardless of whether those feelings are reasonable or justified. They expect that our lives must be organised around managing their concerns. We are still a dangerous anomaly that must be managed, medicalized and (where possible) corrected. It's slightly nicer than talking about children being sacrificed by the trans cult, but not by much.
The self-identiifcation model of gender recognition does exist, it has existed for half a decade in Ireland. It also exists in the Netherlands, in Argentina, in Denmark, Malta and Columbia. There is no evidence that, in any of these places, a self-identification model has posed a risk to cis women. There is no evidence of the kind of mass abuse predicted by TERFs and the "gender criticals". Why would there be?
I think you might not quite be understanding what the argument is here. It is not that trans women will abuse women, it is that men pretending to be trans women will be more able to access female-only spaces and abuse women. And I totally believe there are some abusive men who'll do that. I'm pretty sure some already have.
I had a person I used to call a friend who got into this whole "gender critical" "protect women's spaces" thing. They are a lesbian ciswoman who is butch. Short hair, muscles, always wears men's clothing, but still recognisably AFAB.
Some time after I cut ties with this person, I noticed a facebook post they'd made about being asked to leave a women's bathroom and how awful and humiliating it was, and much as I've had similar experiences and it is really awful I found myself thinking "what exactly do you expect?"
How exactly do you protect women's spaces from trans people? Do you have to carry a special ID card which shows your sex at birth and your current sex? How much detail should it include about your medical history? Should it describe specific surgeries you've had or medication you may be on? Should an attendant be allowed to check your genitals? What if you are allowed to be there, but someone else misidentifies your gender or feels threatened by you anyway? Can they call someone and have you escorted out just because they feel like you shouldn't be there? What if the person they call doesn't believe you are the sex you say you are? Again, do you have to show them your genitals, or your birth certificate, or specific medical information? Can you subsequently sue them for not letting you use facilities you were legally entitled to use? Are you entitled to privacy regarding the information you share to prove you are entitled to be in these spaces? Can this information be shared with third parties without your consent?
I'm pretty sure if you asked the woman who asked my former friend to leave the bathroom, she would say "well, I thought they were a man and I was concerned. I didn't feel safe with them around". Is that a valid concern? Is it less valid because it was directed at a cis woman, even though she can easily say that she didn't know that this person was a cis woman?
Any logistical challenges which people imagine might arise from a self-identification model already exist. We still have to figure out when it is reasonable for a person to feel threatened, or when it is reasonable to ask someone to leave a particular space for the safety of others, and there are still going to be people who feel unsafe sharing facilities with other people even though they are legally entitled to be there.