You mentioned the Kingfisher-like call and the call of the Common Sandpiper does sound a bit like that.
Another clue to this bird’s ID is the fluttering flight you described so well. When disturbed Common Sandpipers will fly away low over the water with a stiff, bat-like wing action.
Common Sandpipers don’t necessarily need mud. They can often be seen tottering around the edges of reservoirs on concrete dams, or on wooden jetties as well as grassy banks. They tend not to wade but feed along the very edge of the water.
Access to the water’s edge and muddy areas can create an unexpected habitat for waders when water is low, but sadly people and dogs also feel the need to explore it and birds taking advantage very often get disturbed.