Many smiths jump to unnecessary and destructive weld repairs on corners when they do not need to. First, anvil corners do not need to be sharp. It is bad for both the work and the tool.
Radiusing corners often clears up minor chipping. More severe chipping can be dressed out by grinding the side of the anvil first slightly narrowing the face. A small amount taken off the side can reduce chips to where radiusing the corners will clean them up entirely. This can also reduce the radius of heavily rounded corners, again, without welding. A few manufacturers ground or machined the side of the anvil just enough to make a true straight corner. So a little grinding there would not be out of place.