The adversary system violates the Establishment Clause. It comes from the disputation method of reaching some answer in Scholasticism. OK in 1270 AD, out of the question as an investigatory method today.
1) Imagine a Koran based jurisprudence. I read their hornbook. 90% of it is pretty good. Still, it would be appalling and unlawful to have legal procedures taken from there. There is a lot of church in the court, the architecture, the robes, the gavel, the high bench, the oaths, the standing upon judge entering the room, the pews, the stentorian tones, the high ceilings, the dressing up as if going to church on Sunday.
2) They got an answer, alright, but 100% of the time, it was incorrect by today's standards. There are no validity statistics to support this bonehead method. There are not even reliability (repeatability) statistics to support it.
3) Twelve strangers, using their gut feelings to detect the truth, will detect only likability, of the defendant, the witnesses, the lawyers, the judge's expressions of preference.
4) Rarity. Because over 95% of cases are settled by an agreement, the trial is a privilege of the rich. The trial, in practice, violates the Equal Protection Clause of the Fifth Amendment for everyone coerced into a plea agreement.
5) To avoid the expense of a trial, the prosecution will use progressively coercive and intimidating tactics to get a plea agreement, even one completely unrelated to the original charge. Such coercion violates the presumption of innocence and subjects the defendant to extra-judicial punishment.
6) The person with the most experience in the court is the judge. He is prohibited from investigating for himself. Should he drive to the scene of the crime, just to get an idea, he will be impeached. This is so the lawyer gets its trial billing. The trial is therefore to generate income, not to arrive at any truth. The argument against inquisitorial judges is that they are biased in favor of the prosecution. They will relate better to police of their race, age, and sex. If evidence of such bias can be found, the judge should be impeached.
7) The jury brings the wisdom of the crowd. This benefit is completely negated by 1) excluding any juror with knowledge; 2) allowing open voting and debate after the first secret ballot, all subsequent votes reflecting the decision of the juror who is loudest, and the rest wanting only to go home. There should be only one secret jury vote.