Unfortunately DXF is not just DXF.. depending on what DXF Visio speaks and what DXF KiCAD understands you can get all kinds of problems.
To make my point here is the Autodesk Inventor Professional DXF export dialog:
So even the 'creators' of the format acknowledge at least 6 versions.
And I can adjust a lot more than just the format version as you can see on the tabs marked.
Any chance you can play with the Visio DXF export settings?
PS: 2004 DXF works for me (Inventor <> KiCAD), didn't try others really.