I have been around pigeons for years--I have never seen one eat an insect, let alone a worm. Pigeons are basically grain and berry feeders. They will strip an elderberry tree of its fruit, and flock to a ripe wheat field but they will not eat your bees.
Sparrows, Thrushes, and Starlings, however, eat what they can find, worms, fruit berries, seeds, insects, etc. I don't even worry about the swallows, they mostly spend their time flying over the water retention pond from the neighborhood next door eating the misquitoes that hatch out of the stagnant water.