You've clearly never trained a dog to a high degree. Yes, dogs can be trained to call off MID chase, people do it ALL of the time, in many different settings. Here's an example in a quick search, and plenty of people teach it with live prey, even with all positive methods!People who think they can train the instinct out of their dogs are expecting too much from them and putting their bird's life at stake. And let's not even talk about cats and birds! The animal world isn't Noah's Ark.
For reference, I have a dog that is HIGHLY HIGHLY prey driven toward my horses. But if we are doing anything even remotely related to bite sport games, even RIGHT next to the horses, the horses are invisible, literally, she doesn't even look at them. That is the level of drive these dogs have toward this game. Literally no prey will ever compete with it, this is the MOST difficult possible call off you can do with this type of dog.
But it should also NEVER come to that. That's where management comes in. Allowing them the opportunity is the problem. You are REALLY focused on the training aspect of it, but that is only a small part of responsibly managing multiple animals (of any species, dogs kill each OTHER all of the time - even though the owners KNEW they didn't get along. which is a rant in itself) in the home.