The four year old should be screened for autism. A 2-3+ year delay in social behaviour is the norm for kids on the spectrum. My son has ASD and a lot of the things you mention are symptoms. Anxiety issues are also very common for autistic kids.
For kids with autism (likely other conditions like FAS too) Alberta there is funding for 'shadow' placements because a child with autism is undoubtedly WAY more work than other kids. This funding wouldn't be in place until a diagnosis (so caregivers get paid for two children when caring for one autistic child).
For you, though, it sounds like it's causing you major anxiety. I would explain this to the parent - write notes so that your don't forget anything (such documentation would help get a diagnosis process started) - reiterate repeatedly that you love them, but that their boys behaviour is burning you out (and you think it may not be his fault - there may be something undiagnosed). You will need to find another family.
Parents tend to freak out with the possibility of a diagnosis - but you can let them know that if there is something more going on, wouldn't they want strategies and support to help their boy thrive wherever he is? For me, I realized we were punishing my son for things he had little or no control over. Being able to tackle behaviours with an assortment of specific strategies is a lifesaver - and the punishment and guilt is over.