'Use Depth texture' was set to false, but setting it to true caused the effect to disappear altogether. That was happening because my level was positioned inside a sphere mesh which I was using for a sky. The sun shafts were being blocked by this mesh.
The fix for that was to add a second camera to my player. The main camera clear flags were set to Depth Only and I put the sphere mesh "sky" in its own layer which I then removed from the culling mask. The second camera clear flags were left as default, and the culling mask was set to render nothing but the sky mesh. I then changed the Depth setting for the Main Camera to 1.
FYI- If you want to be able to see the sun shaft effect through windows or other transparent material you will have to use this same approach, otherwise the sun shafts will by overdrawn but the mesh for your window.