We currently have many such "trials", not by a jury and not under any presumption of innocence. All of the various regulatory boards and commissions, whose members are usually political appointees, preside over "hearings". It brings to mind the clause in the Declaration of Independence "He has sent swarms of officers, to eat out our sustaining."
Another such "trial" or "hearing" is in the realm of family law. Your children can be taken from you by cps, with only a subjective suspicion of abuse, place in foster care (ultimately for adoption) and you have no legal recourse. Both you and your child have been deprived of your constitutional rights. There is no court to appeal to, you can't sue the state or county agency that took them, even if it turns out you were innocent.
Also, tax law.
Yeah, here in CA we have the unelected “Coastal Commission.”
I’ve snickered more than a few times when some poor liberal businessman has had a run in with those people. The CA Coastal Commission is slowly creating more and more Conservatives, due to their arrogance.