So what you are suggesting is a permanent date for winter vacation, assuming it would be on the secular calendar, so weather wise and light and dark wise, it would at the same time each year, say the last 2 weeks of December. The problem with that is that it would make us seem that we are in fact having school holidays to co-incide with Xmas, which is what is done throguhout the Christian world.
No, I don't think so. Let's keep it on Chanukah,

it could be a permanent date using the Hebrew calendar - say 10 Teves for 3 days (or for however long). I picked 10 Teves randomly. pick 15 Teves or 1 SHvat or whatever.

This is the deal - we simply do not prepare 'gedolim'. We second guess everyone now, to the point that we don't believe anyone is righteous enough to lead us. Clearly, some families have prepared their sons (and daughters) for royalty and leadership positions. Yosef, Lau, Eliyahu, etc... Reading about the heads of the family, in the many books on their lives, you see how these tzaddikim grew up, and how their children saw this from the inside - stuff you cannot possibly learn in school, on the street, or wasting time surfing the internet like most of us do.

Claiming there is a fiefdom is truly chilul Hashem of ignorance. If you have better candidates, let's hear about them.

Let's not fool ourselves.
A) The essence of getting elected or promoted to any position is popularity, politics, being well known, and of course being worthy (knowledge and talent) to be a chief rabbi of city.
B) Elections of city rabbis are different than elections of city rabbis and involve a variety of people on the local level from politicians to the shul committees.
C) Father rabbis can simply not appoint their sons to any position, but they certainly might be able to gauge which city might be more welcome to accepting their son's and then have them run there for election.
D) Soldiers must work their way up the ladder of rank in order to get to various positions. Some warriors in the past were able to skip positions and jump higher, perhaps not with a father in a higher rank, but certainly with a friendly general with family/friend relations. Becoming chief of staff definitely has to do with who you know and who upstairs is pulling strings for you.
E) Over the course of Jewish history, many a rabbi have shown at an early stage they are more worthy than older rabbis so comparing chief rabbi to chief of staff is not relevant.
F) Rav David Lau had actually been a candidate for Chief Rabbi of Ariel. That city missed out big time.
G) Politics means working hard all the time and not just at the last minute. Rav Shmuel Eliyahu faced an uncooperative and unprepared Bayit Yehudi as well as one of the great tzaddiks of our time (Rav Yosef's zk"l brachot for supporters). I'm sure if Rav Mordechai had been alive, his son's luck might have been different.
H) I hear you about not willing to run if there is a chance in losing. In order to avoid this, we must demand that the age limit is removed, and start preparing better candidates who we can unite behind. and yes, we must be ready to make 'deals'.