In most Sidurim there is a Yehi Ratzon at the bottom of Refaeinu that one can add in to pray for one who is sick. I have never heard a Chazzan say it during the Chazaras Hashatz. Suppose a Chazan wants to pray for Shul's list of Cholim. Would he be allowed to say it during the loud Shemona Esrei? If yes then why is it not done? If not what is the reason?
First of all, I would think that this would fall under the general טרחא דצבורא category - a Chazzan does not wait for people taking too long to daven or to respond amen to continue with chazzaras hashatz (O.C. 124:3 Rama), so I assume adding in a litany of cholim would also be included.
That being said, the rules of adding (even in one's personal Shemoneh Esrei) would be somewhat restrictive to the Yehi Ratzon as we have it-
If the sick people on the list are not his "own" sick (חולה בתוך ביתו), rather he is praying for כל ישראל, then it must be said in לשון רבים, plural communal tense, and must be said at the end of the blessing (O.C. 119:1 quoting Rabbeinu Yona), as opposed to the Yehi Ratzon as we have it.