I'm constantly astounded that people on the spectrum assume that they're absolutely, 100% right, and that the problem is always everyone else. If I'm saying something, and no one around me is understanding what I'm saying, then the problem is clearly not everyone else. The very clear, and obvious problem is that I'm not communicating clearly -or- effectively.
More often than not, I find that I've omitted something that seems blindingly, patently obvious to me, but no one else was aware of because I entirely failed to communicate it.
This is a hallmark of being on the spectrum; people think that because they see things one way, everyone else must be able to see the same thing.
That one 'blindingly obvious' thing is key often for me too. Sometimes it's not only not obvious to other people but it's entirely wrong too.
Ironically, it's often the same thing the other way round: the neurotypical leaves off or implies some context that seems obvious to them and the people they normally communicate with.
The other main thing, from neurodivergent to neurotypical, is (not) implying emotional meaning. (And vice versa, not picking up on it.) You say something and mean it logically, but hidden in your words is emotional meaning - sometimes it's real but you wouldn't even know it yourself; sometimes it's not real just you said things in a way that someone else would if they meant that extra emotion. Communication is about emotion as much as facts, and the listener rightly tries to pick up on emotions, but misunderstands.
As I said in another comment, one of the defining characteristics of the autism spectrum is a blunted sense of empathy. As you say, that blunted empathy can mean that the autistic person doesn't hear the emotional content, reacts to it inappropriately, or is not able to effectively communicate emotional content themselves.
Come to think of it, if people on the spectrum aren't communicating emotional content, or are doing it very poorly, that might explain part of why some autistic people think they're communicating precisely with carefully chosen words, but their intent and meaning is still being misunderstood.
I'd like to add to that, that 'blunted empathy' can sound like you're a bad person: but as I understand it, and as I think I experience it, it's more about empathy in the precise sense: of not instinctively feeling how someone else feels about something. You can still care about a person, with kindness, but lack empathy. That can lead to not caring and being only self-interested, but doesn't have to. Sometimes the neurodivergent person is trying to express - and enact - kindness, but, being weak in empathy, has to find a different way to express it, one that would make sense to themselves. Often one with specific, logical meaning of words.
So this is kind of a message to neurodivergents. You can be realistic about a relative jack of empathy, but don't need to feel put down as a bad person, or that you're unkind. And I'm truly sorry for all the times people won't understand your kindness because of the 'language' barrier. Wish I could give you all a hug (if hugs are your thing).
Keep trying :-) And I hope there will be some people in your life who will keep trying to understand and express things your way too.
My SO frequently includes me in conversations that they've already started in their head, and I have to remind them that I have zero context for what they just said.
I think probably all people dismiss what is obvious to them as not needing to be said, and for good reason: why overburden a conversation with obvious truths. Though given that we're all just apes with a superiority complex, we're probably entirely wrong about what's obvious or true 🙈
It's a little more complicated with autism though, because one of the hallmarks of autism is blunted empathy (and no, I'm not saying that we're all sociopaths-lite).
An example I heard from a psychologist--and I'm going to try not to butcher this--is that if you show an autistic child a cookie tin and ask them what they think is in the tin, they'll say cookies. Then you show them what's in the tin, and it's actually toy cars. But if, after showing them toy cars in a cookie tin, you ask them what another person is going to think is in the cookie tin, the autistic child is likely to say "toy cars".
Obvs. most people on the spectrum get better about this as they get older and learn from experience, but I strongly suspect that this sort of thing is what's going on when autistic people 'explain' things. My guess is that this difficulty with affective and cognitive empathy is also what leads to people on the spectrum over-explaining things; since they're not able to make an accurate guess about what other people know or can infer, they give too much information about a thing.
Reminds me of that xkcd comic with two experts talking about how people not in their field would only know what they consider basic but people usually don't know that either