A half hour walk (and the half hour back with heavy shopping) would be beyond the capacity of many people living in chronic poverty. I think you underestimate the effects of depression, anxiety, conditions associated with isolation and mental health issues like CFS/fibromyalgia, and the difficulty of dragging reluctant offspring out. Good physical health and energy are not universal, sadly.
This.
That's the thing about privilege. It's often invisible to those who have it.
A half an hour walk into town is a much easier thing for someone who has other people to look after the kids, decent all-weather outdoor clothing, lives in a well lit, nice area with proper pavements all the way to the shops and no need to walk through dodgy areas and has good physical and mental health. You can say - "what's wrong with a half hour walk, I can do it, why can't others?"
But, a mother living in poverty is less likely to have someone to look after the DC while they shop, may well not have proper waterproof clothing for her and all the DC suitable for a long walk in the rain or snow, the journey to the supermarket might go through dodgy areas, or not be on roads made for pedestrians, and they are much more likely than those with money to have physical or mental health conditions.
Your experiences of life, if you're comfortable, are very different to people in poverty. You don't realise how much your privilege smooths the way for you, without you even having to think about it. And that's the real privilege of having money. When you're well off you have choices available to you. You have much more agency. You can chose to do things on your own terms, in a way that suits you.
If you don't have money for long periods of time, then your choices can be very limited and there are constant barriers to overcome and problems to solve that could be made to go away if only you had some money to chuck at it.
We're not talking about half an hour to the supermarket as a one off. You're talking about doing this a few times a week and carrying all the stuff back, on top of whatever other demands the mother has. If she also works when the kids are at school, then your talking about dragging the DC to the supermarket with her, taking up a huge amount of family time, come rain or shine, all year round.
That's not realistic. DC can't spend their childhood being dragged to the supermarket and back constantly. When are they meant to do their homework? Or play?