Gender critical pov is never in favour of either conversion therapy nor gender stereotyping, you seem really confused on this point.
There has been an argument about whether or not permitting any discussion with young people presenting as trans except "affirmation" counts as conversion therapy - TRAs seem to say it does.
Conversion therapy, in this context, is a form of therapy which seeks to convert a trans person into a non trans person. To cure them of being trans in effect. Opposing this does not mean opposing all discussion about the nature, reason for or intended outcomes of a gender dysphoric patient, it means opposing therapies which set out to prevent them being trans or which oppose a trans identity. If a patient presents as trans, at any age, it means that a therapist doesn't immediately say you're probably just gay or a lesbian really, or tells them they aren't trans, or attempts to persuade them they aren't trans, or shouldn't be trans. It doesn't mean they can't have wide ranging discussions about the narure of that, it doesn't mean they should push for treatment, and also, and most importantly it means should the patient's identity change then the therapist should be equally supportive of that. Their identity, and how they experience it, lies with the patient, not the therapist's opinions or ideology.
I've posted examples of what gender identity conversion therapy has looked like in the past.. There are no established medical protocols or published literature that supports this or any other attempt to change a trans identity. And those who have had these practices inflicted on them have testified as to the harm done.
The claim that this will lead to gay and lesbian or gender nonconforming kids being transed is complete bullshit. Contrary to claims LGB people are being erased there are more young people identifying as LGB than at any point in written history. Like much of GC ideology, these claims are based on something that could possibly happen, but which is highly unlikely to and which hasn't happened so far. Almost every aspect of the GC argument is speculative, and not just that but always speculates on the worst possible outcome, no matter how implausible or unsupported by what has gone before. So men will start pretending to be trans to abuse women, LGB kids will all be transed, trans inclusive women's shortlists will end with boards of 50% trans 50% men, no cis woman will ever win another gold medal, all of the trans kids being treated will live lives full of anguish or regret over their mutilated bodies, the census results will be meaningless, there will be no point doing gender pay auditing and ultimately that women will be erased or replaced by as little as using language which is inclusive of trans men in some healthcare settings. All of these claims are complete bullshit. None of them comes with any evidence and in fact in countries which have been more trans inclusive none have happened. It's a bit like when someone claims that accepting refuges means we'll be inundated by undercover suicide bombers intent on destroying the nation, or that any mild concessions to Islam such as serving halal food means sharia law and Islamification of the UK is now inevitable.
That doesn't mean there is no debate to be had, but it should not be one based on ever more lurid scare mongering. Gender neutral toilets for example would not mean the complete erasure of women and toilets turned into dens of predators but that we'll be a bit more like somewhere like Sweden. Some people might not like that and fair enough, let's talk about that, let's talk to the VAWG sector, and the workers and residents, and have reasoned discussions amongst those it affects in sport but not base every conversation on the threat of imminent dystopia caused by a predatory 'ideology' that is seeking to destroy women and children and gays and lesbians and everything that is sacred and good (like prisons and the Olympics). Because that's actually a pretty offensive position to begin with from the trans perspective which is why a lot of trans people do not choose to engage in this debate.
Finally we can only speculate why some on the left have been seduced by such spurious arguments that we would recognise as disingenuous bigotry if applied to any other group. But it seems no coincidence that these attitudes are far more prevalent amongst those of us who grew up in a period when trans people were virtually invisible in public life and usually presented in culture as either the punchline to a joke, a highly sexualised victim, or a serial killer. A generation that despite a few subcultures was raised in an environment in which gender nonconformity was viewed with suspicion - particularly any form of femininity in those born physically male which was policed with derision and even violence. A deeply transphobic enviroment in other words and I find it hard to believe that has had no bearing on the attitudes of some involved in this debate.