I have Games but they lay light to dark cream and one lays a pinkish brown like a BR.
Around here I've only had 2 Game hens I've had a problem with. Caught one and sold her at an auction. Without her backup it put the one I have left at almost the bottom of the pecking order. As wild as the Leghorns I had were I can imagine them mixed with a Game. My Game roos have always been very nice birds, but I wouldn't let them out with another roo. Although Games are said to be tough I can tell you, my girls don't really like to be picked up, but neither do most of my BCMs and I'm down to one Silkie hen and she's a nut case. Try to pick her up and she runs like i have an axe. But the Games get along with everyone. I have some that I am putting my Wheaten Marans over, for the same reason you mentioned, broodiness. I figure with that mix the extra roos go to camp.
Around here I've only had 2 Game hens I've had a problem with. Caught one and sold her at an auction. Without her backup it put the one I have left at almost the bottom of the pecking order. As wild as the Leghorns I had were I can imagine them mixed with a Game. My Game roos have always been very nice birds, but I wouldn't let them out with another roo. Although Games are said to be tough I can tell you, my girls don't really like to be picked up, but neither do most of my BCMs and I'm down to one Silkie hen and she's a nut case. Try to pick her up and she runs like i have an axe. But the Games get along with everyone. I have some that I am putting my Wheaten Marans over, for the same reason you mentioned, broodiness. I figure with that mix the extra roos go to camp.