John: Yes, you are missing something. You've joined the discussion after the end of a very long exploration of just this topic that extended over several years. We've beaten this topic to death.
First, the vibration problem... isn't. You'll just have to take our collective word on that. Scores, if not hundreds, of gyros have deeply immersed HS's, as do such warhorse airplanes as the Seabee, Cessna Skymaster and a few dozen other models... plus the type-certificated production gyros such as the McCulloch J2 and Air and Space 18A. It isn't difficult to make the structure adequately stiff to handle the propwash turbulence without a problem.
Second, it's an easy mistake to make to assume that the direction of the propwash is unrelated to the direction of the freestream. In fact, the propwash is a horizontal tornado that "bends" in sympathy with changes in direction of the surrounding flow. Later, I'll describe a simple experiment to prove this, in case you might still be skeptical.
Third, the propwash does rotate (although it's not quite the tight swirling corkscrew that some people envision). The HS receives this swirling air... and straightens it out. The reaction on the HS produced by this "straightening" process is a torque on the airframe that COUNTERS the torque produced by the engine. As a result, a large, centered HS helps to reduce torque roll and yaw effects. When combined with a full-span vertical fin-rudder as seen on the Dominator, Sparrowhawk and many other gyros, the tail essentially eliminates these effects. This dramatically reduces pilot workload.
Now, the experiment for the skeptics. Obtain two electric fans. One can be a big box fan; the other should be small enough for you to lift and move about easily. Tie some yarn tufts to the big fan's grille at various locations and turn it on. The tufts will stream out, of course, and they'll also lean a little in the direction of the fan blades' rotation, showing you that the slipstream does rotate. Now for the cool part. Take the smaller fan in hand, turn it on and aim ITS slipstream into the INTAKE side of the big fan. Move and tip it in any direction you like -- you'll see the tufts on the EXIT side of the big fan move in response!
What this demonstrates is that the orientation of the outflow is a function of the orientation of the inflow. IOW, the prop slipstream's direction follows the direction of the surrounding flow. Even if you're not relying on the HS to counteract engine-related pitching effects, the HS is still more effective when inside the propwash than outside it (the lift of an aerodynamic surface varies as the SQUARE of its airspeed).
The main fly in the ointment with this setup is that the HS loses power when you throttle back or turn off the engine. You feel this in the controls as an increasing mushiness or "rubbery" feel when doing idle-power landing approaches.