Peter Theobald
Unlike a pheasant, which prefers to spend most of its time on the ground, and has therefore built up strong leg muscles to enable it to run from danger, a pigeon prefers to fly when threatened.

Subsequently, there is little meat worth bothering about on the legs.

However, there is a case to be made for boiling the whole carcass to act as stock when cooking, or stripping the meat off to use in casseroles and the like. Delicious!