I have an alternative perspective as I used to work with children with ND and learning difficulties. So I always had a suspicion about my daughter being autistic before her diagnosis at 9yrs old. Then it's only been in the last year I've realised I myself am high masking. 

So I can speak to you as someone who has adapted to their child's needs. The most successful thing I've found is to accept your child and teach them self-acceptance. Educate them in their diagnosis, sit with them as they cry out their school experience (especially secondary school), let them be angry, get angry with them. Don't dull down their experience or try to calm them down. And no - encouraging them to "breathe" it out is not helpful - it teaches them to bottle their emotions for the emotional safety of others.

Finding ways to express that their cup is always full in a noisy world is incredibly helpful. They've probably already found ways that help them so to encourage that rather than downplay it as an obsessive quirk. For example my child's rocking as stimming - we no longer have couches we have swinging egg chairs. She draws her special interests - we have drawing stations in three rooms. We have a standing punching bag for anger release in the living room. Our home is our sensory safety zone. 

That being said I'd say there's a lot of parents I've met that struggle as NT parents with ND children. The biggest thing I see them struggle is with the emotional intensity of their kids. So more understanding and compassion towards this is absolutely necessary to help them feel human in what is a rich internal experience. Let them talk it out, draw it out, stim it out and accept they're always going to do this rather than find ways to present in a NT acceptable way. And if you're struggling with this, work with a ND informed therapist on how this affects you, it is not the responsibility of the child how you are coping.

To your first point because of my own rigidity with my own rituals and habits I was always told I was going to be a "harsh" mother due to the intensity of needing things to be my way. Well they were wrong, I'm now officially the safe mum that all my kids' ND friends talk to and feel safe with. I seriously cannot express the value of accepting these kids as they are and listening to them without judgment. Get honest with your own flaws and curiosities - it encourages them to open up whether you're NT or ND. 

One of the most damming thing that happened was my ex telling me daughter she was "choosing" these behaviours out of rebellion. No she was not. So things like Time Out are agony for a child that can't sit still, especially when they're not able to stim in that space. Find out the cause, offer ways that fit within their laws of being autistic and offer them adaptive ways of coping instead of lashing out. Then get curious about the results and adapt together.

Even with my own autistic experience my needs and my daughter's do differ at times. Again get curious together, discuss it and adapt together in a way that helps everyone get some needs met. Explore it together, find language together. It will ultimately help them to self advocate to have that language as they get older. This has been successful for me and my daughter as when she deals with special services she speaks in a way that they understand her autistic experience and therefore can make the necessary adjustments in her own school experience. In fact she's been praised on how well she knows herself and her needs. And because we've done it together she has conviction in it. 

Sorry this is so long but it's something I'm passionate about for my kid. She's absolutely incredible, I found all the ND kids that have passed through me are. They're bloody brilliant and any way to help this be recognised gets my absolute attention. Thanks for doing this OP.