I don't know if you consider "games" to be full-sized fowl. They were the backbone of my father's flock and he relied on them hatching and raising chicks. No incubator for him, just the games.
Just because a breed is known to go broody doesn't mean every hen of that breed will. Even a lot of the games Dad relied on never went broody. But enough did that we had to break several from being broody every year. The best "breed" I've had to go broody were Black Australorp from Cackle Hatchery but my sample size was small, just two hens. Both went broody.
While breed is important, I think "line" is more important. Broodiness is something that can be bred into or out of a flock. If you never hatch eggs from hens that go broody in a very few generations you will have a flock that does not have hens going broody, even Silkies. On the other hand, if you keep breeding chicks from eggs laid by a hen that went broody, broodiness can greatly increase in your flock.
I hatched eggs laid by those "broody" Australorp crossed with other breeds. A fair portion of the pullets went broody but certainly not all of them. Then I saved a rooster from one of those eggs. In a couple of generations practically every hen went broody, either as a pullet or a year old hen. The genetics for "broodiness" can come from both sides.
If you can get chickens (hens, chicks, or hatching eggs) from a flock that regularly has broody hens I'd consider that a good starting point. Or when you get a broody hen, hatch her eggs and keep the chicks as replacements.
