The son of the Kohen Gadol (if he is worthy) takes precedence over any other Kohen to succeed his father. Two Kohanim actually served in the positions of Kohen Gadol – the regular High Priest – and Kohen Mashuach Milchamah – a Kohen Gadol ordained specifically prior to the nation’s entrance into a war. This latter one was a specially designated Kohen Gadol whose purpose it was to address the nation and give his charge before the battle, encouraging them that Hashem will protect them. The Kohen Mashuach Milchamah is forbidden to marry a widow and may pose questions to the Urim v’Tumim. The function of the Kohen Mashuach Milchamah was just that: to address the nation prior to war. Such a Kohen could go through life with the august title and never do anything but speak publicly one time. One distinction between the Kohen Gadol and Kohen Mashuach Milchamah is inheritance. The Kohen Gadol’s son succeeds his father; the Kohen Mashuach Milchamah’s son does not.
Returning to our opening pasuk, Chazal (Yoma 73a) teach that the rule that the Kohen Mashuach Milchamah’s son does not inherit his father’s portion is derived from the pasuk that adds: “Who shall enter the Ohel Moed to serve in the Sanctuary?” – only the son of a Kohen Gadol who enters the Holy of Holies may inherit his father’s position. The son of a Kohen who only performs the duty of addressing the nation prior to war does not inherit his father’s position.
Having said this, we cite a well-known teshuvah, responsa, from the Techeiles Mordechai, Horav Mordechai Yohlin, zl (served as Rav in a suburb of Kiev, Ukraine, then emigrated to America, where he was Rav in Philadelphia. Niftar Erev Yom Kippur 1942). He was asked about the laws of chazakah, whereby a chazzan had a long-standing position leading the Shacharis service during the Yamim Noraim. After a number of years, an assistant chazzan was appointed, who, due to the fact that the primary chazzan was still active, would instead lead the services on Shabbos and Yom Tov. Since he did not work during the Yamim Noraim, he took a position elsewhere during these days.
The question was: The first chazzan left this world for his eternal rest. His son claimed chazakah (the halachic status of permanence that is established when an event repeats itself three times), since his father had held the position of chazzan for many years. The assistant chazzan claimed that, by right, it belonged to him as the next in line. The only reason he was not present for the Yamim Noraim was that he had no work. Furthermore, the rule of chazakah should not apply if it is not consecutive, 365 days a year, for three years. The late chazzan had only worked three days a year!
He quotes Horav Moshe Nosson HaLevi Rubinstein, zl, who cites Chazal in Meseches Yoma who apply our opening pasuk as support to disallow the son of the Mashuach Milchamah from inheriting his father’s position. He suggests that if, in fact, a chazakah of three incomplete times was invalid, why would Chazal require a pasuk to teach that the Mashuach Milchamah’s position does not go to his son? The Kohen Mashuach did his service from time to time. Thus, the chazakah was faulty. Apparently, the idea that Chazal found it necessary to employ a lesson from the pasuk to invalidate the son of the Mashuach Milchamah is an indication that otherwise he would have succeeded his father, based on the rule of chazakah.
I suggest that the chazzan, who led service three times a year, and the Kohen Mashuach Milchamah, who performed his function sparingly, are both entitled to the benefits of the rule of chazakah. How often a person carries out his function has no bearing on his title or position. At the end of the day, he is the shul’s chazzan, and he is the Kohen Gadol Mashuach Milchamah. This title is ongoing; thus, it provides him with a chazakah.