This doesn't pass the sniff test to me, at least for most COL levels. The only way that our household was able to reduce our food budget by eating out was to massively reduce quality in order to meet caloric needs. For roughly the equivalent quality of ingredients it was much cheaper for us to cook at home. And that's before we got a bigger house in a lower COL area with a chest freezer, which allows us to bank high quality ingredients when they're on sale or buy in bulk, especially for meat. I could probably eat cheaper still at McDonald's, but at the cost of my health.
Subjectively, I don't think that cooking at home is a "luxury". Maybe being a foodie is an upper class thing given the extra cost and time investment associated with it, but historically huge chunks of the middle class have cooked at home for a large chunk of their meals. Generally speaking only the lowest economic strata of the working poor depend heavily on fast & packaged food, and that's largely a factor of both time and ingredient availability. If you're working two or three jobs or live in a food desert, you're not going to cook at home no matter what the unit economics are.
That being said, I do have a general theory about high COL businesses. After moving out of our high COL city (Santa Monica, for the record), we've started building out a home gym and programming our own workouts. What we realized is that while our old gym provided a lot of knowledge and social structure, the primary service they offered was land and equipment. They were able to charge a premium because we literally couldn't afford enough land to put the equipment. Now that we live in a lower COL city with a large garage, the financial balance points towards it being cheaper to just buy all the equipment rather than paying for a gym membership. At the extreme end of land cost, I can see this even applying to kitchens too, since a good kitchen takes up a fairly large portion of an apartment.