According to Shia school of thought. Those incidents you mentioned happened numerous times. Though they happened to show Ali's virtue.
Usually it was like Umar/Abubakr made a judgement but then Ali was either there to correct them or they didn't know what to do so they asked him or the someone informed Ali to come and help.
few people claimed that a women has committed adultery with a non-Muslim and and the women had a husband and she should have been stoned.
The woman said : Oh Allah you know that I'm innocent
Omar became angry and said do you deny?
Imam Ali who was present there said, maybe this lady had to commit adultery.
So they brought her back and asked her.
The lady said: One day me and my neighbor who was a man took our camels to eat and because of that my camels didn't have milk I took some water to drink. But during our travel my water finished and I was thirsty. I asked the man to give me water but he denied me unless I commit adultery with him. I didn't want to do such but I was about to die from thirst so unwillingly I submitted myself to him.
Imam Ali said: Allah Akbar :
فمن اضطر غير باغ ولا عاد فلا اثم عليه
But whoever is forced [by necessity], neither desiring [it] nor transgressing [its limit], there is no sin upon him
Upon hearing such Omar let her go
A man who had drank wine was taken to Abubkar, so they would punish him. Abubakr asked: Have drank wine? Yes! he said.
Abubakr asked: Why did you drink it while it's haram? "I am a Muslim but my house is far away from here where people deem wine as halal and I was never made aware of it being haram and had I knew I would have never drank.
"What should we do about this?" Abubakr said to Umar.
Umar said the answer is with Ali. Abubakr told his servant to go get Ali. Umar said we shall go to him. They all went to Imam Ali, Salman Farsi was also there. Salman explained the story to Imam Ali. Imam Ali told Abubakr, send a man with him to go to all the houses of the Muhajir and Ansar and if anyone has read the verse of Tahrim Sharab then they shall be a witness but if not then there is no punishment for him. Abubakr commanded as such but no one said he had been informed so he was freed.
Salman said: You guided these people. Imam Ali said:
افمن يهدى الى الحق ان يتبع امن لايهدى الا يهدى فمالكم كيف تحكمون
Say, " Allah guides to the truth. So is He who guides to the truth more worthy to be followed or he who guides not unless he is guided? Then what is [wrong] with you - how do you judge?"
So having that said, the incidents you refer to where a result of something else.
Now about the system of correction. During the caliphe of Imam Ali:
Whenever he assigned a governor, he sent a letter to the governor informing him of his tasks and morals of being a governor AND he also sent a letter to the people informing them of their governor and his duties. He had no tolerance for hypocrisy. Yet he is bound to govern people by the seen not the unseen ie if one person is a hypocrite he can't punish him. He can only punish him for the actions that have witnesses.
Whenever a person made an unintentional or intentional error Imam Ali was swift to take action.
Either the person would approve of his mistake or if not then the person usually fleed. The structure was as such as if there was a person making wrong decisions all the time then the people would inform their caliphe and he would take action. Or if the governor didn't know how to judge something then he would not take action.