Very interesting comment, Dickeybird. I too, figured that a thicker fin would develop higher lift. I built a small scale fuselage and fin/rudder assembly, and wind tunnel tested it with both a thick, and a thin fin. The test was simple. I measured only the angle to which the rudder would deflect the fuselage, relative to the wind. Tried a relatively thick fin, carved from foam, and then, a thin, sheet balsa fin. Result: The thin fin produced identical yaw angle, within the limits of experimental error. Rudder area was equal to fin area, and deflection 45 degrees. I have used relatively thin fins since then, although I usually build them up with ribs and spars, to give them an airfoil thickness, including the rudder, of 5% or so - makes them stiffer, and thus more resistant to flutter.

Some quite serious full-scale airplanes, such as the Extras and early Edges flew quite well with flat-plate stabs. A flat plate develops a steeper slope of lift coefficient versus angle of attack than does a thicker section, so that in theory, a flat plate stab is a slightly more effective stabilizer. I use flat plate tail feathers only on rather small, slow models, since they are far more prone to flutter, requiring bracing at higher airspeeds. Even with bracing, their thin, flexible elevators are more prone to flutter than the thicker elevators used with a thicker stab. It might be interesting to ask Edge 540 pilots whether they noticed any improvement in pitch control when the airfoiled, thicker, unbraced stab was adopted.
I like NACA 009 or 0010 sections for horizontal stabs.