While I will concede that clerical trials might be marginally better in the eyes of some than the secular courts of the time, the key word is "marginally", and even then it's unclear how often a church tribunal would be better.
The main issue was that if, say, a lettered man murdered someone in cold blood in broad daylight, a clerical trial would involve him going somewhere else, and lose the immediacy of justice that the medieval mind understood. The punishment might be less severe, and in any case THERE WAS NO DEATH PENALTY IN CHURCH TRIALS.
While the justice system was notoriously unreliable and involved setting up a "hue and cry", finding a suspect and then attempting to figure out if he was indeed the culprit, you can see how this could lead to a certain sense of impunity among those who were lettered and could try to claim the benefits accruing to the clergy. This was a serious problem.
Additionally, because trial by ordeal was the standard for many crimes, a clerical trial which involved the "ordeal" of taking a Communion wafer as opposed to ordeal by water or glowing red iron, made innocence easier to prove. In Henry II's time we aren't talking about a system even as developed as what you're imagining.
Not true. It worked in France and Spain.
ἡ δὲ κἀκ τριῶν τρυπημάτων ἐργαζομένη ἐνεκάλει τῇ φύσει, δυσφορουμένη, ὅτι δὴ μὴ καὶ τοὺς τιτθοὺς αὐτῇ εὐρύτερον ἢ νῦν εἰσι τρυπώη, ὅπως καὶ ἄλλην ἐνταῦθα μίξιν ἐπιτεχνᾶσθαι δυνατὴ εἴη. – Procopius
Ummaka qinnassa nīk!
*MySmiley*
