I didn't have the patience to read through the article about bird's vision. Still I don't think the head bobbing has anything to do with vision. If they moved their heads from side to side, that might be different.

Walking is not a pigeon's preferred method of transport. Flying is. I think their bodies are not well adapted for walking, and bobbing may restore balance as they waddle toward a goal.

Or it might be that a pigeon spies a crumb, and the small brain directs the small head to bob toward the crumb. The message from the head progresses down the spinal chord toward the feet, which finally get the idea and commence waddling.