The wall sits on the wood which sits on a rough 2" concrete block.
How should I cover this edge so that I have a single clean surface from the floor to the wall?
In all cases like this, I have this covered with a durable paintable material. In your case I would use a 1X PVC material to cover from the floor to the top of the 2X plate and another piece of PVC or 1X wood to cover the top of the plate and PVC edge and paint it all.