This is a funny aspect of English. In general, we distinguish the meat of mammals (beef, pork, venison, mutton, ...), but non-mammals are their meat (fish, chicken, ...).

But I don't think this is well explained by a class distinction. I mean, the classes of people who raised sheep weren't so different from the people who raised chickens.

chicken, duck, goose, pheasant, squab / poultry?

One of those doesn't belong before the /:

Pheasant: from Old French fesan. (the current french is faisan).