As far as I understand, a person has the right to a "fair trial" in most countries. This is considered a human right and a constitutional right in many countries. Now in relation to this, there are different types of trial systems. 2 of them are jury trials and bench trials. In my country is always being used a jury trial. Now in some cities/provinces, it's being optional to have a bench trial. This could become mandatory someday. In my personal opinion, this type of system which is used in countries like United States I believe, it's flawed and unfair. In my opinion, common people dont have the knowledge to make a fair judgement and most of them judge according to prejudices and beliefs. Apart of this, when you take a bunch of random people to judge, who are never going to judge again, and which they dont have judging as a source of labor income, they dont have a reason to be responsable other than their own moral which in some people is unexistent, they arent in the spot if they take a bad decisition because they are part of a group of several people and they havent been formed to judge, they dont have a job to lose, and they dont have a record of bad jugdments if they are bad at judging, so bad people judging arent removed of the system as bad jugdes in jury trials are. So my question is, have ever a trial in the world have been appealed and succesfully reverted the judicial decisition, because of the court system used being considered "unfair"?
While the answer from @K-C is correct as far as it goes, the decision regarding whether a jury trial is or is not a fair way to go about making decisions, which is the type of issue that you single out, is a political one that cannot be used be individuals to invalidate a jury's decision.
Ultimately, the large scale pros and cons analysis that you conduct in your post is a policy decision that courts are not free to hold is invalid.
While you are of the personal opinion that a jury trial is unfair, your view is not widely shared in places where jury trials are used as an important part of decision making.
If you don't like jury trials, the only way to change that is to change the law and possibly the constitution, before you are involved in one.
Yes, an unfair trial is at the root of a lot of appeals: ineffective assistance of council, closed court during voir dire, Batson challenges (relating to racially motivated veto of a jury member), long delays before trial, being denied counsel of choice, etc.
Except for structural errors, the defendant must show that the error caused them prejudice. For structural errors, prejudice is deemed to have occurred. Especially for structural errors, the question is whether the error has rendered a trial "fundamentally unfair".