I can see flaws in this logic, just looking back to my younger days of being part of the working poor. Even then, I knew that home cooking was cheaper and healthier than fast food. The reason I did not cook that much at home came down to a few things. First, I lacked many of the basic utensils one needs for cooking. I had a few hand-me-down things my folks had given me when I moved out, but that did not include good knives or sauce pans. Another problem was that food does not keep forever, and when I went shopping for home-cooking ingredients, it often went bad before I could use it, and I could ill afford to spend money on food that went to waste. At least with fast food, there was no waste. Finally, I didn't know how to cook.
In my case, at least I had the luxury of having the time to cook. I couldn't do it well (I still cringe at the meal I once made of, what I charitably call, unleavened bread topped with ketchup), but at least I had the luxury. In fact my problem was that I was underemployed back then. I think in that respect I was better off than people trying to survive by holding down multiple jobs.
Today, many of the working poor are working two or even three - often physically demanding - jobs to make ends meet. While there may be time to grab a burger between jobs, there is hardly to come home and cook a proper meal. This is assuming that the person even as the energy left for that at the end of the week.
That still does not stop well-meaning, but slightly arrogant and clueless folks going on at length about how cheap things like rice and bulk meats can be. One lad went on about how there was always time to cook a healthy meal at home. You just needed to go out and buy a 10-pound bag of rice, some super cheap bulk meat, and frozen veggies. You just throw the rice in your rice cooker, toss the meat in a pot and stir it a few times over the heat, and then throw the veggies in the microwave. He priced it out to $150 a month to eat this way.
We'll skip over the part about this desperately poor person investing in a rice cooker and a microwave - though the latter makes a bit more sense - and skip to the meal itself. For $150 a month, this person could eat a meal every single day of plain rice, plain meat stirred up in a pot, and reheated vegetables. Every day. It's wonderful how the privileged can find such simple solutions for these stupid poor people. For a mere $5 a day, they can have the same bland meal day after day after day.
Or, for about the same amount, and with virtually no time investment they could have a much tastier burger and fries, or a taco, or any of the other variety of fairly inexpensive fast foods. Even better, they can grab it between jobs so that when they get home they have time to sleep and not worry about cooking and clean-up.