Emily - there must be lots of tasty grubs in the bark, which the birds are finding irresistible. It's a good thing too, as these aphids, grubs and moth caterpillars could potentially become pests of your fruit crops. Nature has a great way of keeping pests in check so I would let the birds get on with it.
I'm not sure the birds would be after your bulbs - squirrels certainly, but I've never come across bulb-munching birds. Perhaps the birds are also after soil-dwelling creatures under your shrubs, such as vine weevils, leatherjackets, chafer grubs and worms etc. The (now resident) blackbird in my garden sifts through all my pots, hopefully keeping vine weevil populations in check. I don't mind having to repot the odd misplaced plant if it means the birds get to eat and the pests get eaten!
Kate