"when I think of the sort of jobs that it's acceptable to advertise at NMW, I think of jobs that literally anyone with the physical ability could do. Things like stacking shelves and using a coffee machine"
That may be what you think of WooWoo, but that doesn't reflect the range and disparity of minimum wage jobs at all. Lots of people in caring professions and education are paid minimum wage.
I also think it's quite disparaging to say that "literally anyone with the physical ability" could work in retail or hospitality. Lots of these jobs are incredibly physically demanding when performed over years and decades (much more so than most office workers imagine) and also require great customer service skills - which NOT everybody has, and involves internalising a lot of stress.
I disagree that "the problem isn't NMW in itself, it's that there are jobs that are wrongly given NMW when they shouldn't be". Lots of posters here want to argue that companies will only pay what the market will bear - so why should those free market principles apply to retail and hospitality staff, but not to carers and educators. If people want to blame low pay on supply and demand, then there's no wriggling out of it just for the jobs which you personally happen to sympathise with.
The fact is that the minimum wage is too low for anyone to live on adequately while working full time.
Nor do I agree that "nothing will change until there is a labour shortage". There is lots that the government/employers/civil society can do to improve things for low paid workers. Just introducing the minimum wage was a start. Sadly it was worth more in real terms when it was introduced than it is today, in 2014. It hasn't kept pace with inflation, and that really penalises people on very low incomes when the cost of living has shot so high.