Firstly, Mazal Tov on the new home! It is special that you made a Chanukat Habayit at the time of the mezuzah placement. In this way, you are dedicating the house to its higher purpose.
Your rabbi was referring to the Sages instruction that “It is greater to do one’s mitzvah by himself than to appoint an agent.” The authorities write that this is applicable to mitzvot in general and to mezuzah, in particular. We find that the great Sages were careful to perform their mitzvot themselves, even when this entailed menial and arduous labor.
In contrast, the placing of mezuzot on the doors of the Jewish Old Age Home was a communal and not a personal obligation. Therefore, there was no issue in distributing the honors among the community leaders.
Even so, many authorities record that it is a common custom for individuals to forgo their rights in deference to a revered rabbi or other community leaders worthy of distinction. Your father would certainly fit into this category. Perhaps your trusted friends could be included as well, if your intention is to honor their loyalty. This is why your rabbi acceded to your desire in the end.
Your rabbi made sure to tell you to at least start off the mitzvah in order to minimally fulfill the concept of doing the mitzvah on your own. He instructed you to preferably appoint the others as your agents for the same reason. As for having them listen to your berachah, this was in order that your berachah should count for them.
 Targum Yonason to Devarim 20:5 translates: “He who has built a house but has not dedicated it…” as “has not affixed its mezuzot…”. See introduction to Chovas HaTalmidim, who cites Rashi’s definition of “chinuch”: the dedication of an object to its designated purpose. In this sense, the affixing of mezuzot dedicates the home as an abode for the Divine Presence.
 Kiddushin 41a.
 Magen Avraham 250:2 and Mishnah Berurah 250:3.
 Tevuas Shor 28:14; Binas Adam, Sha’ar Issur V’heter 7.
 Kiddushin 41a.
 Be’er Moshe (2:95, cited in Sha’arei HaMezuzah 16:2) opines that the director of the institution may make the berachah as the representative of the community.
 See Pitchei Teshuvah O.C. 658:12.
 Shiurei Berachah Y.D. 264:1.
 Bei’ur Halachah 135, cited in Agur B’ohalecha 8:17:39.
 Agur B’ohalecha 8:17:40.
 Da’as Kedoshim 289:2.