Converts are not "מקרב אחיך" and thus are not allowed to be dayanim, in my understanding. Why then are they not מקרב אחיך? Thanks in advance.
From perusing the multitude of writings on that verse (Devorim17:15) it seems that Chazal learned this Halacha from the fact that it didn't use a more popular term for "all Jews" like כל האזרח בישראל.
From the fact that the Torah uses the term "from amidst your brothers" it seems to be stressing the fact that he really has to be a blood relative, not just a Halachic relative.
The Gemara in Kidushin (76b) already learns that this verse applies to all positions of power/authority and not only to being appointed the king.