Having four cats myself. It can be done I assure you.
My two youngest cats are the only issue.. my older cats.. have not now nor ever been serious "hunters".
By that I mean they might chase a bug here and there.. but don't kill said bug.. in fact if the bug makes a move.. they retreat.
My younger boys however.. have posed serious stress... on myself and my birds.
I had to anchor the cage, that was my biggest issue. The young ones could topple it over.. so now it's encased in a heavy metal shelving unit.

The young ones have gradually lost interest. But here's how I cope.
I have the cage near the computer .. tv area.. which in fact is the social area of our home.
The cage remains open as long as I'm in the room now.
I keep a water bottle near me.
I do not think it's impossible to train a cat to not bother a bird.
I do not think you can ever completely trust your cat won't bother your bird.
If you love the cats and birds.. it can be done... Just takes some work.. it's way harder if you have an actual "hunting" cat though.
o_0