Not that I disagree in principle, but the way you say this suggests that otherwise the literate class should be tried by nobles selected by an accident of birth or a jury of illiterates. Henry II was, at best, just getting the English justice system off the ground. Asking the Church to surrender its own protections and trust this relatively new justice system is going to work out, is a bit much. Our experience with the civil courts is based on the finished product of centuries of evolution and adjustments. In the 12th century, the choices were the king's justice, which was really a power grab against the nobles (how'd the end result of the centralizing of royal power against the aristocracy work out for the Church when it hit its apex four hundred years later? ), or the local aristocracy's whim. For an educated professional, a trial by the Church, which was far and away the closest thing to a meritocracy among the major institutions of the medieval world, was probably a better bet. Down the road, if you were going to be accused of witchcraft, you'd be much better off being accused in Spain under the Inquisition, which threw out thousands of cases, than in England or Germany.

The failure was that the Church didn't have an Inquisition or any means of going after them, and supine clergy, fearing the bad opinion of the world more than God's Justice, kept it quiet (and weak parents took payouts to leave the next kids vulnerable).
I don't see where that worked anywhere else. Once a Church gets nationalized, that's the inevitable end.