I feel like we probably need to address the fact that, unfortunately, the rigid fix rules based world views of society are inherently appealing to those on the spectrum.
I really really don't think this is true at all, even though I agree that autistic people can be bigoted/fascistic (just like there are members of any other minority group that can do this). The whole "rules" thing, in my experience as an autistic person, is more a consequence of wanting to know how things are likely to be ahead of time (predictability, rather than valuing of "rules" on the basis of authority imposing them and wanting to submit to some authority - though of course that can also be the case but its not an autism thing and you see it in non-autistic people as well). This is in combination with some people viewing "the rules" as a determination of fairness (which is also common in non autistic people nya).
This can appeal to some people's ideas on ethics if they base them more on external structure rather than their own ideas of ethics. Which frankly seems rarer to me in autistic people than in non autistic people, though its still common.
Not that I'm saying autistic people can't be bigots or fascists and it may well attach itself to some of these other concepts, but I really don't think it appeals to autistic people as a whole more, whatsoever.
Combined with the fact that autistic people are very often treated horrendously by people around them due to arbitrary "social rules" and societal structures that punish people for being outside of the norm, and I think this is more likely to drive people away from supporting any system that is pro conformity (such as fascism). The other autistic people I talk with and am friends with (which is most of my friends xD nya) would suggest this, but of course that's a very biased sample.
Furthermore, a larger proportion of autistic people call themselves lgbtq+. Theres a good chance a lot of this is related to social factors preventing non-autistic people from examining themselves or coming out or similar, but it remains a true effect on whether on not someone considers themselves lgbtq+. And I suspect that that itself acts as a strong push for more autistic people to oppose fascist and other conservative ideologies.
Definitely I used to be a big """rules follower""" when I was younger, but it didn't take me very long to realise that just because following "the rules" made certain things more predictable/cognitively-easy did not mean that the rules themselves or the system behind them was good in an ethical sense. Its made easier when rules hurt you or other people and when you can see the direct similarity between neurotypical social rules (which i often find ridiculous and I can't actually even try-and-fail to smulate without it making me very depressed) and systemic rules.
I think the actual main danger of falling to the fascists for autistic people is similar to how it is for non-autistic people - social isolation and the false acceptance of fascist spaces like much of 4chan, usually layered in enough irony that you can pretend they don't hate autistic people and anyone different from societal norms because they also act like they hate themselves (which they may actually do - this also happens with trans people on certain 4chan boards nya) - often the people getting involved with this hate themselves for being autistic and view themselves as being inferior (or sometimes you get the autistic supremacy people or people who do the "ironic" superiority thing when they actually hate themselves).
This can then of course act to make people "learn" bigotry via social normalisation and unexamined talking points which are often made up and always misleading, combined with the need for going along with it for pseudo-social-acceptance.
Personally I've seen that a lot of autistic people are capable of deconstructing this stuff (if raised with it) when pointed to enough contradictory information, talking to people of different perspectives (as you've experienced yourself), and/or also given acceptance via other groups. Or just finding the internal contradictions all on their own, which I know more than one person who has done that without even needing much external input. But again, I have a biased sample in who I interact with.
I'm trans nonbinary, and I get very strongly bothered (not in this case as you are asking but...) when people suggest that being nonbinary means you aren't trans (or that they are mutually exclusive groups), as my gender is distinct from that which was assigned to me at birth nya.
I share experiences with a lot of other trans people as well and so having people imply that I am separate from that community bothers me.
This combines with the fact that such separation often implies that nonbinary people want or need only a subset of things as binary trans people may often want/need (such as various aspects of medical transition), which is untrue - for example, I have HRT, much like many (but not all) binary trans people want or have, but often people separating "nonbinary" from "trans" seem to operate with the implicit subtext that nonbinary people do not transition or do not transition in ways they consider as "real" or "full" or "as important" as binary transition (not that I am suggesting that is actually a thing, just illustrating the subtextual framing I perceive - its also worth remembering that binary transition pathways/goals/etc. vary a lot between people too).
Not only this but this separation often comes with the implicit subtext/assumption that nonbinary identities are not strongly separated from their assigned-at-birth gender identity. There are certainly nonbinary people like that, but there are also lots of us who have an identity quite far from that assigned at birth ^.^.
Some nonbinary people don't consider themselves trans, but a lot of us do (including me).
In summary, for me this is because my gender identity (strong sense of gender but not one of the "man/woman" genders and instead a third axis with some fem component) is strongly distinct from my AGAB, and I do/want-to-do things that other people in the trans community often-but-not-always do or want to do (such as accessing HRT, legal name change, surgeries (though not conventional binary ones), etc.)
As for the etymology, just think of it as cis=same, trans=different - rather than cis=same, trans=opposite.