Someone can be exhausted with one child, they have other things going on, work, perhaps caring for a parent, perhaps financial stress, perhaps chronic illness, or just exhaustion, she doesn't have to justify her exhaustion or meet a certain threshold of life situations to be allowed to say she's exhausted, t's not a competition. OP also said the has 4 kids.
Plus, OP has a kid with SEN which is likely more exhausting than a parent of multiple kids with no SEN. He can't just be better, he needs more support (from school and other resources), the reason she wants to take him out is to a safer place, that makes sense. She doesn't want him to get stuff from the bin etc, OP would have to only be watching him if she let him play in the garden. He likely has issues with impulsiveness and also is having lots of incidents where he's running around the house or banning into things, so getting him out to something like soft play could help to get all his energy out but in a safe environment.
It's not the same as kids who don't have those needs, but don't have a garden, I understand what you're saying those families manage, but her kid has different, more present needs. Plus it doesn't mean those families wouldn't like that, or wouldn't benefit from that, even if they 'manage' it might not be managing that well.
This has nothing to do with his families financial situation or being in a family with neglect, abuse, poverty, mental health etc. So no telling him kids go through that won't do anything, he already has challenges, (yes he could have others, but again it's not a race to the bottom, we don't have to only care when a kid has SEN and is neglected and lives in poverty), the comparison isn't helpful. He can't help it (that's why he needs extra support to learn emotional regulation) and that takes more time in kids with his issues and also needs specialist support, different types of therapy @FatAmy123, maybe consider play therapy? He also need support in school, as OP said he's not at his actual age, and even if he was, all 7 year old haven't learnt complete emotional regulation yet. It's not just that, he clearly isn't safe on his own, trying to leave the house or screaming and breaking things. Yet the council have let her down, when normal these kids can get support like TAs with more training in school, DBT or play therapy, sometimes respite care so parents can have a break, they often need more accommodations in school, perhaps a special school for secondary @FatAmy123 because they'll likely have the facilities for him.
Boundaries and consequences will help but don't work the same for kids with SEN. He doesn't have the best understanding of those concepts yet. Tell me you don't understand SEN conditions without telling me. Many professionals are often needed to support these kids, it takes longer for them as well to do so, expectations need to change as a result to be fair for them, yes they need to understand behavioural rules but that's the longer term goal and we need to adjust expectations in the short-term to get them there.