The primary reason that a Ger may not hold a position of serara, is because he does not have zechus avos. If the people over whom he has serara are moser din lashomayim on him, he will need zechus avos to protect himself.
The gemara in Brachos 27b mentions this as a reason that R’ Akiva did not get the job of nasi when Rabban Gamliel was deposed.