I had duck tonight, Chinese take-out, and sorry to say but I should have cooked my own. Undercooked... greasy. Duck can be greasy but properly cooked it doesn't have to be, shouldn't be.
Lamb is one of my favorite meats. I don't understand why so many people don't like lamb. Lamb too can be greasy if not properly cooked, but again it doesn't have to be. You just have to understand that both lamb and duck can be greasy and you need to take proper steps to cook them and end up with a result that isn't greasy.
I've had deer (venison) only a few times but really liked it. I often wonder if people get some nursery book idea that cute little duckies, lambies and "Bambi" aren't for eating. On the other hand I'm not very fond of eating insects and other garden critters. Perhaps it depends on what you were fed by your parents when you were young. My mother served lamb often, usually lamb chops.
The venison I had was just the opposite of duck/lamb, very un-greasy, very lean. I suspect that leanness is a challenge to cooking venison properly, although I've never cooked it, only had it cooked for me, just speculating.
Perhaps people don't like duck and lamb because they had improperly cooked greasy duck or lamb and were turned off by the grease (that shouldn't have been there in a properly cooked dish). I attribute dislike of deer to the "Bambi" effect, and to large city populations who are not exposed to game meats, didn't have it when they were young, think it's some kind of weird food to eat.