What room do you keep your birds in ? Is there a smaller room with a door you can close that you can use to train the birds to fly around in. We try to exercise our bird in the bathroom where it is a smaller space and easier to catch. I know how moms can be since I am one too. If she says the birds need to stay in the cage then that is what you have to do. Maybe you can try to finger train the birds while they are in the cage and then show her what they can do?
Popcornbird, just like some cats are indoors only and others are indoor/outdoor, some birds will be ok if they are safer in their cage (like if there are also cats and dogs in the house). Your birds are very lucky to have you as their owner and have so much freedom.