Alas, I suffer from more than cockroaches in these parts - we have an active and voracious fox population which takes the lambs given half a chance, and the rabbits breed like, er, rabbits if left unchecked.
I leave the cockroaches to the cats (who clearly have no taste. Ugh!)
I agree that ducks aren't pests (I just like mallard and teal and my local butcher charges a fortune for them) - though Canada geese are (and also pretty tasty, to boot). Deer in the UK need management because they have to fit into man-made territories, and the population's self-management strategy in the face of that involves starvation and a lingering death for the less fit specimens.
I agree, though, that 'trophy hunting' is pathetic.