The Beit Yosef[2] cites the Maharik, who maintains that since the prayers correspond to the temidin,[3] which must be funded by the community, a chazan must be acceptable to everyone in the community. The logic is that it is improper to appoint a shaliach (agent) who is unwanted. Based on this, the Shulchan Aruch[4] rules that an individual’s protest against a potential chazan is accepted, as long as it is made prior to the chazan’s appointment. The Rama[5] adds that this refers to a case in which the protestor can convince the communal leadership that he has sufficient cause and rules that hatred between people qualifies as sufficient cause.[6] Thus, your inclination to disallow the functionary in question ostensibly has a basis.
However, Acharonim distinguish between the aforementioned sources and the most common modern applications. The Magen Avraham[7] and Mishna Berura[8] write that the Maharik’s logic applies only when one needs to rely on the chazan to fulfill his own prayer obligation, i.e., when people would listen to the chazan’s repetition of Shemoneh Esrei instead of reciting it themselves. However, nowadays, the chazan only leads the congregation in certain sections and provides cantorial flourishes for parts of tefilla, while each person himself recites all that he needs to. Consequently, we revert to the regular rule that the majority has the authority to make appointments to various communal tasks, and individuals lack veto power.
Realize also that many of the qualifications for a chazan (found in Orach Chayim 53) concern the position of the shul’s main chazan.[9] In that case, the most appropriate person should be chosen, which may exclude one with any serious blemish. The Aruch HaShulchan[10] indicates that animosity is reason to invalidate only such a permanent chazan, but not one who will be serving sporadically. As he mentions, we cannot possibly tolerate an “endless” process of disqualifications.
Irrespective of the fact that not everyone needs or deserves the special privilege of being the chazan, the issue of embarrassing someone and depriving him of the honor of occasionally being the chazan should not be taken lightly. Almost all those who are capable of leading services have done so at some time. Although a particular chazan may be a flawed individual (which we cannot judge from afar), many flawed people lead services. That some individual congregant may be unable to concentrate is not a significant factor. For any number of reasons, one may be bothered by another person’s davening, and it is not feasible to grant veto power purely on subjective grounds.
On the other hand, when it comes to a chazan for the tefillot of Rosh Hashana and Yom Kippur, we do seek a higher level of qualification, and his relationships with members of the community play more of a role.[11] This is especially true regarding blowing the shofar on Rosh Hashana, which resembles the classic case in which an individual can object. The shofar blower enables others to fulfill their mitzva of hearing the sounds of the shofar, not everyone receives the honor of being the ba’al tokeiah, and the claim can be made that each year is considered a new appointment.[12] Therefore, if there are viable options to replace him with someone who is similarly capable and gets along with everyone, it is reasonable to alert the local rabbi/leadership. They would have to weigh the obviously not black-and-white matter on its merits.[13] However, we cannot stress enough that it is much more appropriate that the effort be made to reduce tension and defuse the issue than to replace the person and undoubtedly raise the conflict to new, more regrettable levels.