But again, you are completely ignoring any form of lateral shift. Nobody has to progress in their own job, they can simply move sideways into an industry which is either relatively better paid or has better opportunity for progression overall.
In your example of working in a care home, one can move into NHS as an HCA or, better still, a rehab assistant. RAs are typically band 4 with zero managerial responsibilities. That's already above NMW.
Having gained some experience as RA, one could consider a degree in OT, PT or SaLT. Maths skills are not required for any of these roles. After graduating, you start on B5 which is a bump in salary. Realistic progression from B5 to B7 is about 2-3 years for an average grad (area-dependent). B7 is roughly double of NMW.
Talking to people is greatly valued in many professions, including the ones I mentioned above. But one could also study to become a counsellor. They could stop there which is already above NWM of, again, consider a progression to assistant psychologist and all the way through to clinical psychologist. Great opportunities for private work there as well which takes one well above NHS pay grades.
In your example of ASD friend struggling to work in a supermarket - there are quite a few options of non-customer facing roles in other industries. These range from anything like cleaning jobs (a lot paid better than NMW) all the way to cyber-security, game design etc.
My aunt was left a single mother with three young kids when she was working in retail. She signed up to a bookeeping course which allowed her to take on private contracts. Some years later she completed a degree in accounting and she's been working from home for close to 20 years now. My friend with severely disabled child did exactly the same thing.
Your friend who couldn't share accommodation due to trauma could have transferred to an OU course part time.
Some creative thinking goes a long way.