I agree that it's hard to eat well on a budget, but having said that, the choices do not have to be between organic "artisanal" foods and totally processed canned foods. It's a false dichotomy. There are options that fall along that spectrum of extremes.
I am not feeding a family, but I am still feeding myself on a tight budget. I never buy canned beans, for example. A 1-lb bag of dried chick peas costs less than $2.00 and takes just a few hours to cook in a slow cooker (while I'm not even paying attention - just dump the beans, water, some salt and seasonings into the crock pot, press start, and leave it alone while it cooks). Minimal investment of both time and money, and it helps feed me for the entire week. Or I'll spend a few hours on a Sunday cooking, for example, a huge pot of pasta sauce or soup. I leave out what I plan to eat for the next week or so and freeze the rest. Yes, it takes some time on that day, but after that, the time it takes to reheat something for the next meal is the same amount of time it would take to reheat a can of pasta.
That's not the whole story, of course, because I'm certainly not eating only chick peas or reheated pasta for the whole week, but the point is that eating healthier can be done on a budget (both time and money) without having to shop at the fancy health food stores or spend hours cooking every day.