Dayan Goy

A mistake is forbidden because the Gentiles are in order and the reason for the law is nullified.
And why should the judge not nullify the law due to the nullification of the reason? (Al-Goi 3:8 and the question is correct)

I guess this enthusiasm is misplaced. You probably meant to comment on the thread about the Akko error. I'll move it there and I'll answer there. Maybe you can cool off a bit along the way. 🙂

https://mikyab.net/%D7%A9%D7%95%D7%AA/%d7%a7%d7%91%d7%9c%d7%aa-%d7%9b%d7%9e%d7%95%d7%aa-%d7%92%d7%93%d7%95%d7%9c%d7%94-%d7%99%d7%95%d7%aa%d7%a8-%d7%9e%d7%a1%d7%9a-%d7%94%d7%94%d7%96%d7%9e%d7%a0%d7%94-%d7%a9%d7%91%d7%95%d7%a6%d7%a2%d7%94#comment-56428

Now a clarification has been made there, and it became clear to me that the question was actually directed here. And here is the clarification:

The intention of the contradiction is different and the comment is in its place. There the rabbi said that a Gentile is disqualified from discussing, and did not say that in their day, when Gentiles were not bound by the manners of the nations, they were disqualified from discussing, but today a Gentile scholar can discuss just as a Jewish scholar can. And in this he found a contradiction, a contradiction in the words of the rabbi who, regarding the error of the Akkam, decided that it was precisely in their time that the gentiles were not circumcised. The question is how do you decide what law applies to a gentile and is also valid in our time and what law applies to his behavior and is not valid in our time.

My response:
A strange question (and therefore I did not believe it belonged here). The gentile's disqualification is not due to his dimensions or behavior. He is disqualified because his body is disqualified from judging and testifying. Therefore, the Meiri's opinion regarding the kashrut of judging does not belong. The disqualifications of son-in-laws also did not change following the change in the conduct of the gentiles (it is not permitted to marry a gentile man or woman) and the same applies to the prohibition of enjoyment of those who practice idolatry.
True, regarding courts, we find that they are competent because they are not based on personal reasons, and the first and last ones have extended this law, and so on. In any case, it is clear that this is not useful where a formal judgment is required (as in conversion and the like). And in this regard, we have found In Gittin, when their laws were as ours, the law was pronounced "before them and not before the people of Acre."

Perhaps if their laws are according to our law, it is forbidden to go, but if they are judging according to our law, that is, judging according to the Torah, and even more so if in a Jewish court in the Sanhedrin, then perhaps it is permissible for a Gentile to judge and there are no courts involved and he joins the quorum of the court.

Apparently, according to the version of his rabbi in his book on the subject of bringing up a foreigner, a judge in the Sanhedrin can be one of the descendants of Yitro through the descendants of the Kenites and the Rehabs (also relevant because next week's torah is Yitro):
Rashi on Sanhedrin, page 141
These are the Kenites who come from the father of the house of Rehab – and these Kenites are the sons of Yitro, as it is written, and the sons of Kenites, Moses' father-in-law, and the sons of Yitro are the sons of Rehab, as it is written, the Kenites who come from the father of the house of Rehab and it is written (Jeremiah 3:1) And concerning the house of Rehab Jeremiah said, for thus says the Lord [And] because you have heard [about] the commandments of [Yehudav] your father [and] [Yehudav] that no one should be circumcised for Yehudav the son of Rebekah [standing] before all the days, and we have written them in the books, perhaps they were circumcised and entered the temple, and indeed all Israel did not enter the temple except when they sat in the Sanhedrin and were instructed in the words of Torah, from the mouth of Rabbi: