> Putting money away in savings, hopefully to secure a more stable future, should bring some measure of confidence/happiness.
Only if you actually believe that saving money secures a more stable future. In practice, it really doesn't - you're still working at the same shitty job with no prospects, just with nothing to distract you from the reality of what that means, and maybe if you manage to save enough over half a year you can survive a couple of weeks between jobs. Woop.
It doesn't actually provide stability in the sense that makes people feel happy - a job in which it's unlikely you'll get fired without warning in the first place does that. Hopping jobs, hopping rented accommodation, hopping shitty transport that you can't really depend on but you need to depend on for a living... that's the stressor. Many people with professional jobs, stable housing, and health insurance, but no savings to speak of are quite happy.
You're never going to save enough to actually do anything with it within five years on a low-income job, no matter how much you cut back - doubly so if you have any debt at all - and there's studies that show that planning for the future where that future is not guaranteed is not a strength of many people on low incomes.