Walk has no function. The character either runs or sprints. Walking is not possible. I even tried changing around the key bindings - nada - nothing - zip - zilch!
Mine works fine. Hit the X next to the WALK, then refresh it to the default BACKSPACE and see if that fixes it for you. If that does fix it, then it must have something to do with your choice of binding (Left Shift may not be possible). If it doesn’t fix it, than re-verify game files perhaps?
Edit: The problem may be that walk is a toggle (even though it doesn’t specify that). So holding Left Shift could be problematic, additionally, any press of Left Shift would be constantly toggling the walk on/off for you.
Edit 2: You’re right, if I am holding any weapons (including tools or a torch) than it disables the walk, switching me back to sprint. Definitely feels like a bug, rather than intent.