I agree that it doesn't sound like a cat. I have the neighbors cats plus some strays come through and none show any interest whatever in my orps. Besides, if it's eating the birds like you say then it's pretty doggone hungry, and I doubt a pet would be starving like that. Not to say a cat might not kill a smaller bird, just that I doubt that's what killed the majority of them. The camera sounds like an excellent idea.
But a cat can indeed be kept up. In the house. There is no reason a cat *must* be allowed to roam free. Most cat owners do prefer to have them outside but they can live a jolly good life indoors and never set foot on grass. I love cats and have had them most of my life. I do not currently have one because I don't want one in my house after a room mates cat peed in various places in here. That was 7 years ago and I've treated the spots but once peed at, another will do so without hesitating. And I won't have one that is only or mostly outside. They are marauding bird and wildlife killers and are exposed to too many dangers if allowed to roam free.