I haven't been able to locate a definitive source that authoritatively answers your question, so I hope that you'll be fine with what I infer from this source. The article, and the site is comprehensive, and I haven't found any other that delves into the history of tefilot as well. I will summarize some of the key points in the article.
There is a controversy on the definition of the term הזכרת נשמות - "memorializing the souls" and how that translates into specific prayer forms. Two versions spawned from this.
The version composed by שבולי הלקט (Zedekiah ben Abraham Anav (1210 - ca. 1280)) - states that a prayer should be recited by anyone present in synagogue on Shabbat morning
who wishes to recite a prayer for the benefit of a deceased relative provided that the one reciting the prayer includes a promise of a gift to charity. This version is written in the 1st person and is what we now know ad the "short" יזכור prayer that is recited as part of that service on major Holidays. This is the "silent" prayer recited by individuals - see the linked article for details.)
The other version is מחזור וטרי (composed by Simhah ben Samuel of Vitry, a French Talmudist around 12th century). His versions provides for a format that is recited by the Chazan (he doesn't mention Gabbai, though, I infer that it refers to any congregational "leader" who can do this.) to honor deceased individuals who
have contributed to the benefit of the community either through the study and teaching of Torah or by having bestowed a physical benefit to the synagogue or whose family did so after his death. This version is what is now known as the El Maleh prayer that you are referring to.
Based on the history of the prayers, this would explain why the Chazan or Gabbai is the one reciting it. It was meant as a communal prayer to acknowledge the important individuals who contributed to the shul. As to how and why this "tradition" shifted, i.e. - that emphasis is no longer there, I can't say, and I might be able to locate something about this.
I wouldn't know if an individual can say this prayer, either. I haven't seen it done. I viewed a few web funeral home web sites and similar that discuss gravesite and unveiling prayer rituals. All of them said that "El Maleh" should be recited at the grave only when there is a minyan present. Regardless, now that you understand the history of the prayer, it seems, IMO, inappropriate to recite this prayer individually. Rather, one should recite the "shorter" version.
The tircha detzibur problem - Yes, I see what you mean, as I am a Gabbai, and I have this question often. In terms of obeying the history of the prayer, there's nothing within the article implying that each person needed to be mentioned individually. Look at the article, and you'll see it mentions that the original text was similar to the ending of Yequm Purkan, which mentions no names. So, I infer that even then, there was no concern about grouping names together.
In my shul, we "compromise". Individuals that approach me get individual prayers. We ususally don't get more than 2 per day, anyway, so the extra minute is not a big deal. Those that submit their names are on a list, and I group those together. Regardless, follow your shul's custom.
I hope all this helped. Read the linked article, and I think you'll glean some great insights.