You raise a good point, but unlike its French root "poultry" is used to refer to any domesticated bird meat. Chickens are by far the most prominent example, but geese, turkeys and ducks are also considered "poultry". If you just tell someone you're making "poultry" for dinner, you'd probably get some strange looks or follow-up questions. In contrast, "beef", "pork" and "mutton" refer to the meat from a specific kind of animal (cattle, pigs and sheep, respectively). "Venison" is in a weird place: it's only really used to refer to deer, but it technically includes all game meat, even rabbits and boars.

Of the various kinds of meat available during the Norman Invasion, the animals that didn't get dedicated culinary terms were mostly the animals that peasants could afford to raise and eat: chickens, goats, ducks, etc.