"Autism Speaks" is a US organisation, and I'm not in the US, so the debate over them doesn't have a lot of relevance to me personally. That said, while I disagree with many of their beliefs, and I don't doubt they cause some harm (albeit unintended rather than intentionally), I think a lot of the rhetoric levied against them (such as "exterminationist") is overwrought.
I think a lot of the problem is a problem with "autism"/"ASD" itself – throwing together into one bucket two very different groups of people – one group is people who have average-or-greater IQs (sometimes even with significant giftedness), albeit combined with various social and emotional problems, anxiety, topical obsessions, sensory sensitivities, etc. Many of these people can realistically aim for many of the same goals as the average "neurotypical" person (education, a successful career, a partner/spouse, children, home ownership, etc), even if they are likely to experience certain bumps on the road which the average person might not. Some may benefit from some extra help, but many will achieve a lot even without it. Then there is another group: people with significant, sometimes even profound, intellectual disability; many of whom are non-verbal; some of whom have significant problems with self-injury or physical violence towards their carers. Goals such as a university degree, a professional career, marriage, children, which many people in the first group manage to achieve, are totally unrealistic for people in this second group. We've given both groups of people the same label, "autism", despite the fact they have very little in common and clearly "don't have the same thing". (Of course, there is also a third group of people who don't really belong to either, but are somewhere in-between the two.)
Organisations such as Autism Speaks are dominated by parents of children who belong to the second group, those organisations which vehemently criticise them are run by people who belong to the first, who (in my view) are misinterpreting statements about the other "autism" as statements about their own. And, for all their concern about the harms caused by groups like Autism Speaks, don't want to hear about the very real, albeit somewhat different, harms to which they and their organisations contribute.
> but I know the exact same problem with trans people
I think the parallel you are trying to draw here is very questionable. I'm not aware of any ideological or personal overlap between Autism Speaks and the kinds of social/religious conservative or radical feminist groups which are involved in controversies over trans rights/equality. On the contrary, if you look at Autism Speaks' own website, they present themselves as LGBTQ-friendly: https://www.autismspeaks.org/lgbtq-information-and-resources
> This is also the big reason a lot of autism and trans advocates heavily oppose researching autism and transness within genetics. Because it looks like a precursor to elimination.
That fear seems to me to be grounded in ignorance of the actual state of autism genetics research. For one thing, most researchers don't believe that ASD actually exists at the level of the genome. "ASD" is really a kitchen sink into which we've thrown a whole bunch of unrelated and poorly understood conditions, and the eventual outcome of improving our understanding of the genetics is very likely going to be the abolition of "ASD" and its replacement with a bunch of more specific concepts with a much better genetic basis. Also, the DSM-5 boundaries between different conditions (such as ASD, ADHD, OCD, etc) likely don't exist at the genetic level – the same genetic variant might increase the odds of all three of ASD, ADHD and OCD, and which (if any) of them you actually develop may be determined by interactions with other genes, non-genetic environmental factors (whether biological or psychosocial), or even just random chance. In other words, you can't genetically eliminate ASD, because genetically it doesn't exist to be eliminated.
You might be able to genetically eliminate certain specific forms of syndromic ASD–aka "ASD due to a known genetic condition", to use the terminology of the DSM-5; which most people with ASD don't have, most cases of ASD are non-syndromic: a few non-syndromic cases may actually be undiscovered syndromes but the vast majority likely have highly multifactorial origins which rules them out of the "syndromic" category. Some even argue that syndromic ASD shouldn't be counted as ASD, but rather as the syndrome, and we should only consider non-syndromic ASD to be ASD. If gene therapy could be used to reduce some of the disabling symptoms of Fragile X syndrome (one of the most common types of syndromic ASD–not everyone with FXS meets the diagnostic criteria for ASD, but 15-60% do, with that wide variation in diagnosis rates mostly due to differences in ASD diagnostic practices), such as intellectual disability, seizures and early onset dementia–what's wrong with that? FXS, as one of the best understood genetic causes of intellectual disability, is also the target of significant drug development efforts, which doesn't directly change the genetic mutation, but attempts to counteract its downstream biological consequences. Is it wrong to try to develop drugs to boost the IQ of intellectually disabled people?
Similarly, there appears to be some genetic contribution to transness, but all the research I've seen suggests its genetics are diffuse, and you can't use genetic manipulation to eliminate a genetically diffuse condition–once again, a fear which seems to be grounded in ignorance of the actual state of genetics research. (Also, whether and to what extent transness and homosexuality/bisexuality are distinct at the genetic level is an open research question.)