How did "oyster catchers" ever get that name since mature oysters never need to be caught, they are as stationary as a stone?

The name oystercatcher is actually misleading because these birds rarely eat oysters. The name originated in the 18th century when the English naturalist Mark Catesby observed them along the American coast and assumed that they were mainly oyster-eaters.

In fact, oystercatchers feed mainly on mussels, clams and other bivalves, using their strong, chisel-like bills to pry open or crack open their shells. Although they can eat oysters, they usually choose easier prey such as softer-shelled molluscs or worms found in sand and mudflats.

Before the name oystercatcher became common, these birds were sometimes called sea pies because of their black and white plumage.

https://en.wikipedia.org/wiki/Oystercatcher