In your standard crossflow head you have, essentialy, a bent tube that brings air into the combustion chamber, and its controlled by a valve that drops down and the air slips by. The shap of the top of the valve is actualy good for filling the chamber with a good charge. You can go bigger with the valves (to a point), and you can open them farther (to a point), and you can control when with basic adjustments.
With the above system you are limited to how big a spinning sphere you can fit over the combustion chamber, and how much of it you can clear out to let air by. The inlet/outlet shape itself isn't conducive to smooth flow. On top of that, it would be tougher to get a good seal on a higher compression engine, standard valves are actualy helped shut by compression/combustion, the forces in the above system would be constantly working to launch the sphere into the stratosphere. Plus you cant make simple adjustments. If you want to change something, you need a damn good machine shop.
Also, bearing surfaces. Bearing surface = friction = parcitic loss. Standard valve, you have a bearing on the ends and one between each cylindar, plus the friction of the cam lobes. With the above system, you have a bearing surface of the entirety of the shaft. In the end, its actualy more complicated, more expensive, and less efficient....but it's neato.