I love to cook, and I am pretty good at planning the week's meals for cheap, but I have the luxuries of time, a spouse, and easy transportation. I think it's completely possible to stretch a very thin budget to allow for a healthy diet, and to do it much more inexpensively than a McDonald's diet would cost.
But there are many factors involved. The first is that a some poverty-stricken people are working full time or more than full time, with more than one job, just to get by. Then they have to come home and be parents. I know that if I had only one hour with my kids per day, I probably wouldn't want to spend it cooking.
The other was brought up in this thread already: with poor transportation, many people only have the option of shopping at the local convenience store, where the healthier foods are incredibly overpriced. I have a vehicle. It's old and ugly, but it gets me from point A to B whenever I want it to, and I don't have to pay extra for parking. In the US, good food just doesn't tend to be within walking or biking distance of residences, especially in urban areas.
There's also marital status (or live-in partner). If I can shop while my husband watches the kids, I won't have to hire a babysitter while I go shopping after they're in bed. I won't have to drag them with me, either. Or I can ask my husband to pick things up on the way home.
So, all else being equal, I think it is fairly easy to maintain a healthy or at least semi-healthy diet on a thin budget. However, all else is not equal most of the time. Sure, it's possible, but there's much more to it than just making a meal plan and sticking to it.