Cheap chicken is a very recent thing - like mid 1900s recent. Chickens need to feed on grain (unlike cattle), and chickens also produce eggs (unlike pigs), which means that they're both relatively expensive to raise (before industrial farming made grain cheap as well) and have a huge opportunity cost to slaughter for meat.
Modern chickens have ridiculously bulked up, and their feeding has been highly optimised, with the fastest growing breeds you can turn out a 4lbs broiler in 6 weeks, roman chickens would be far less optimised for meat, and as you note feed grain would be scarcer, so turning out a chicken for meat would also take a lot more time (hence even more expenses).
Similarly, dairy cow had been far less optimised, and so had dairy “husbandry”. Furthermore, cow milk was considered quite inferior to goat or sheep (not to mention ancient rome would have had pretty high levels of lactose intolerance, which are less of an issue with sheep and especially goat milk — a fact sometimes noted by roman writers e.g. Pliny). Thus dairy cow husbandry would have little presence in Roman culture, instead cattle was mostly work (oxen) and meat. Therefore the only balance (and opportunity cost) would be between these two, not unlike horse.
Table scraps, what? When food is half of income, table scraps are much more meager, much less left on the table, and there's a whole pecking order of who eats what cuts of meat (hopefully meat) and everything else. All the way from the breadwinner down to the pig.
Foraging? Yeah if you don't mind foxes eating them, which is like the favoritest thing a fox can want. They figure out chicken coops built with way higher production values with chicken wire and the like, in the present day, back in Rome it must have been a cakewalk. And I don't blame the foxes, nor do I blame native Chilean cats that do the same, rather it's just to easy to prey on this flightless bird humans feed to eat. The whole concept of animal domestication, really, it's fish in a barrel.
Table scraps is not exactly a scalable solution. That's fine for a handful of chickens a particular family owns, but a) those chickens are not going to be slaughtered until they can produce no more eggs and b) they will likely be consumed by the family, not sold on the market as it is just a handful of animals.
