Accepted examples are to prevent serious crop damage and to protect stored grain etc,
I have never understood the crop damage claim with regards to crows. A good 90% of a crow’s diet is carnivorous – insects, nestlings, mice, small birds, frogs, eggs, snails, earthworms, and of course carrion.
A carrion crow is called a carrion crow because it eats carrion, not because it eats wheat, maize, or any other crop.
A crow would very quickly die if it tried to live on a vegetarian diet. Although they do like some fruit as they have a sweet tooth, so to speak. But when they are in fields they are almost certainly searching for insects, grubs, mollusks, etc, not wheat or barley.
I can only imagine that they are beneficial to crops, not detrimental. A crow’s diet is pretty much identical to the natural diet of a fox, I’ve never heard anyone accuse foxes of damaging crops.
And I’ve never been convinced that “scarecrows” work, crows aren’t stupid they can tell the difference between a man and a couple of sticks with clothes on. Although I can see that scarecrows might scare pigeons which do damage crops.