It ain't that simple.
Keeping to some LOOSE similarities with this incident... Let's say a guy is thrown out of a house by the cops after beating his wife half to death. There's a protective order issued against him. Or even just legally evicted from the place for whatever reason... He's still got ID saying he lives there, but he doesn't necessarily have a right to be there. Some extra digging is appropriate even if he's got an ID that lists that address. What about a bad divorce or messy break-up?
Sure, the ID goes a long way towards supporting the claim. But it's not automatically going to end the discussion. A few more minutes chatting, a quick record check... and at least the cop has done what he can to make sure everything is right.
Your right. But they were called because of a break in. When they ascertained that the man with the ID was the man who broke in that should have pretty well finished it. There was no complaint about domestic violence. But you've made some valid points, perhaps a quick records check would have been a good idea.