How can birds disperse seeds when they are able to digest them?
Birds may indeed digest seeds under conditions of rest. It has been postulated that almost all current knowledge on mechanisms of internal seed dispersal has been obtained from experiments with resting animals.

A study with the mallard Anas platyrhynchos (common wild duck), claimed to be quantitatively one of the most important seed dispersing animals in aquatic habitats in the Northern Hemisphere, has shown that physical activity affects gut passage survival and retention time of ingested plant seeds. The authors fed seeds of nine common wetland plants to mallards trained to subsequently swim for six hours. They compared gut passage survival of seeds against a control treatment with resting mallards. 

Intact gut passage of seeds increased significantly with mallard activity (up to 80% in the fastest swimming treatment compared to the control), and hence revealing reduced digestive efficiency due to increased metabolic rates. This enhances the dispersal potential of ingested seeds.

An extensive work-out after a meal generally slows down metabolism, as blood is redistributed to the musculature and away from the digestive tract. 

These findings imply that seed dispersal potential by mallards calculated from other experiments with resting birds may be underestimated.

Reference
- Kleyheeg et al., Oikos (2015); 124(7): 899–907