There have been many attempts at rotary/non-poppet valves in four stroke engines. The Bristol Centaurus, a WW II radial aero engine, was relatively successful using sleeve valves. Super Mag summed up the problems in a nutshell -- rotary valves either run dry and seize up, or burn oil excessively. The Centaurus's sleeves had to be hand-fitted to their respective cylinders and did not interchange, and oil consumption was described as "acceptable", which might be a lot higher than a car engine could live with. On the positive side, the Centaurus was very compact for its displacement and much quieter than a conventional radial.
Maybe the solution lies in materials, graphite or ceramic or....