How I Edit Text That Is Already Baked Into an Image

There's a specific image problem that most tools have no good answer for: the words are already part of the picture, and the original design file is gone. A cafe renames itself, but the sign photo is already printed on the menu. An event moves a week later, but the poster only exists as a JPG. A price goes up and the shelf label is a flat photo from two years ago. In each case, re-shooting costs more than the change is worth.