According to Bird Feeder Hub, these 14 birds eat blackberries:
Red-eyed vireo
American robin
Cedar waxwing
Red-headed woodpecker
Northern mockingbird
European starling
Gray catbird
Bluebirds - eastern, western, and mountain
Brown thrasher
Baltimore orioles
Spotted towhee
Indigo bunting
Northern cardinals
Purple finch
Source: https://birdfeederhub.com/birds-that-eat-blackberries/