If you really have a monster...and I do too, then there isn't much to be done but keep things out of reach. Once I have cleared the house from top to bottom, Charlie - when he is in the mood - will find something. Oven knobs are his favourite but since I generally remember to remove those and put them away these days I'm sort of winning the war if even I've lost many battles. Putting fetch on cue does not stop this behaviour. The criteria of 'behaviour happens in response to a stimulus, and doesn't happen without it' is only relevant within the context of a training session. Just like sit being properly on cue does not mean your dog won't sit without you saying sit. Shake is another example. Shake can be properly on cue, but this doesn't mean your dog won't shake when left to his own devices. Picking stuff up and bringing it to a human can be in the same kind of bucket, outside a training or work session. There is nothing you can do but remove the stuff your dog can pick up. You won't have to do this forever, it is a habit and they do grow out of it (depending on how obsessed they are). You may as well give her treats for handing stuff over, by the way. That will help your 'give' cue, but won't make a jot of difference whether or not she picks stuff up in the first place. The only thing that will stop that is 'leave it' or picking up your stuff. Including your oven knobs.
Ziggys fetish is dish cloths...If I firget to put them out of reach he will sniff them out and steal them off the bench or out of the kitchen sink.