Bill Houghton, Sebastopol, CA
Always better to be safer than you absolutely have to be. Good improvement.

You might look for "pan" or "truss" head screws, ideally in internal hex ("Allen") head. These are even lower profile, and don't have a slot to create an edge.

The next step would be making a recessed slot (wider slot) partway into the fence to receive the screw heads; but this may be more work than value.