A robin's eyes sit on the sides of the head, so it cannot look straight down with both at once. The characteristic hop, freeze and head-cock brings a single eye to bear on a patch of soil. The bird is looking, not listening in the way the folk description implies.
Controlled experiments on foraging robins tested the available cues by suppressing them in turn. Robins located buried prey when visual cues were available and their success dropped when those cues were removed, while masking noise indicated hearing contributes as well. The picture is a bird using more than one channel, with sight doing heavy work rather than the ear alone.
The wider misreading concerns the robin's status as a sign of spring. Many American robins do not leave northern latitudes at all. They stop defending lawn territories, form flocks and switch to fruit, so they become far less visible without going anywhere. The bird reappearing on the lawn in March is often announcing a change of diet, not a return trip.