I think you're right that people can often be wrong about that. Since most people tend to make assumptions based off of appearances, and autistic people often might have problems expressing their emotions or miscommunicate it by not having the correct body language or something, most people might get the wrong impression that the autistic person doesn't feel any empathy or is rude/mean.
I don't know if that applies to absolutely everyone on the Spectrum though, just because I'm not sure any one thing applies to everyone on the Spectrum. Absolutely 100% agree with the point that there is a lot autistic people can contribute if they can overcome their anxieties!
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After years of self-imposed exile. I am now making an effort to talk to people. So anyone feel free to PM me on any subject, I would love to try to interact with people more!