Look again at the wing on the Sumthin' Extra. Now tell me again why you're worried about reducing drag in an airfoiled stabilizer?

As for thick vs thin..... Normally a thicker section tends to stall at a higher angle of attack than a thin one. But with stabilizers and fins it's a different story since they are variable camber sections due to the fixed and moving portions. So the rules the wing follows don't totally apply. The variable camber means that a thicker section isn't as important. This is why a lot of classic planes can happlily get away with flat tail airfoils.

All that being said it's possible that the airfoil nose will help keep the air more attached up to a point but given the deflection angles used in a not of 3D airplanes avoiding a stall is near impossible. But just because the tail sections are stalled in the rear half doesn't mean the tails stop lifting. They still do that very well.