Low earners should have had the sense to realise they were never going to have a life of luxury and always going to struggle financially. They could accept that, or they could take steps to improve their lot, by taking a second job, getting further qualifications, working towards promotions, working longer hours, etc.
We do actually need people doing shit jobs on minimum wage. So even if one individual manages to break free, we'll need someone else to step in to do the job. So there will still be the problem of people who haven't been able to save for pensions.
Likewise almost all company structures are pyyramids - lots of jobs at the bottom, fewer team leaders, even fewer managers. Only a small proportion of those on the lowest levels are needed to fill the level above.
It's very reassuring to say "we don't need to make provision for the poorest, because their poverty is a result of their bad choices", but there will always be people at the lowest levels, that's how our society works.
Incidentally, taking a second job is difficult for those on minimum wage since it is likely to lead to the loss of the first job - not directly, but because the employer requires you to be able to work on any day and at short notice. If you can't, you'll get fewer hours assigned to you, right down to your contract minimum which could be 0.
If people worked closer to full time hours, then that would also help the labour shortages in the sectors you mention. In many industries, employers deliberately have contracts of 20 or fewer hours a week as a way of managing peaks and troughs. Pay as little as possible during quiet times and ramp up the hours hugely during busier times. And I've already explained why it is hard to manage 2 limited hours jobs simultaneously.
What we can do is make pensioners pay more in because of their healthcare costs. A pensioner is simply someone of your age in a few years time. Makes more sense to pay in more when you're in your peak earning years rather than pay more in once your income has dropped to half (or less).