They are Muslim but I didn't want to put that in my original post as it wasn't relevant
It's totally relevant. Everything you said about them struck a chord with me and I knew as soon as I read your OP they were going to be a (probably first generation immigrant) Muslim family.
I was scrolling down the thread to see how long it took for that info to come out. I've lived in an Arab country long enough to recognise certain behaviours as completely and utterly normal among certain cultures and nationalities. Of course this is a cultural thing rather than just a Muslim thing.
Even down to the child throwing food around and the mother ignoring it, and the half hearted 'shush' to their screeching. Completely brattish behaviour from the boy children is like water off a ducks back. No seatbealts, check. Letting the small child wander around unsupervised near moving cars, check. All very normal where I live.
And the reason she probably doesnt take the children out much is that she's probably not allowed or encouraged to leave the house unless the DH is with her., or at least a family chaperone. Also perfectly normal to stay indoors a great deal during the day they drag your toddlers out to a shopping mall or to visit relatives until midnight or later on a week night.
Every single thing that's been described here that is flagging up possible neglect or abuse to some of you wouldn't pass muster where I live.
If the child's constant screaming is down to autism or some other a SN then there may be some cultural resistance to the idea of LDs. The child is made that way due to Gods will and they just work around it without seeking a diagnosis, in fact they will often be quite indignant if you suggest they might like to seek one, and if teachers etc can't cope then that's a failing in them, not an issue with the child.