There are problems with an adversarial system-- attorneys are rewarded for using weak but plausible arguments for their clients, and as stated above, there are also inequities in the quality of attorneys that different people can afford. Further, attorneys are so expensive that poorer people get an attorney supplied by the court who may not bother to try, or be so ill-paid that they can't afford to put a good case together.

There are also problems with an inquisitorial system. Iterated prisoner's dilemma implies that the investigator may be on the side of the court more than on the side of the accused. In theory, juries are supposed to prevent that, but US practice has been to give juries less and less information and to tell them that they have less and less freedom to use their whole minds.

I assume that a Bayesian court would permit-- possibly even encourage-- jurors to do additional research during the trial.

Some of the problems can be ameliorated by combining the systems-- three voices, speaking for the accused, the defense, and a third that's hopefully a neutral investigator.

Permitting the jurors to ask questions (afaik, a relatively recent addition to US practice and not universal here-- I don't know about other countries) is a significant contribution to getting more minds on the problem of getting all the relevant information and arguments on the table.