The fixed calendar already prevents Yom Kippur from falling on a Friday or Sunday, to avoid two consecutive days where all melachot are forbidden. (See Shulchan Aruch Orach Chaim 428:1 with e.g. Magen Avraham, Biur HaGra, Biur Halachah.)
If we were to also prevent Yom Kippur from falling on Shabbat, that would lead to a situation where the new moon of Tishrei could appear on Tuesday night, but Rosh Hashanah wouldn't start until Friday night (a full three days later) in order to avoid Yom Kippur falling on Friday, Shabbat or Sunday. To my mind, the Rabbis would not have wanted to set up the calendar in such a way that would lead to this ridiculous-seeming outcome.
As noted by @DonielF in a comment, there is precedent for this line of thinking in Arachin 8b-9a:
אמר רב הונא לא נראה לחכמים לעבר יתר על שמונה מאי שנא תשעה דלא אם כן קדים אתי סיהרא תלתא יומי
Rav Huna says: It did not seem appropriate to the Sages to extend more than eight months in a year [and establish them as full, thirty-day months].
What is different about nine months [in that the Sages did not deem it appropriate to establish that many full months in one year]? If so [i.e. if there were that many full months] then the new moon [of Tishrei] would arrive three days before [Rosh HaShana].
קדים [אתי] סיהרא - חדתא ג' ימים קודם ראש השנה שהמולד ביום ד' וראש השנה בשבת ומרנני עלמא בתר רבנן למימר כל מה דבעו עבדי
The new moon would arrive three days before Rosh HaShana because the molad will fall on Wedenesday but Rosh HaShana would only fall on Shabbat, and people will mock the Rabbis, saying that they do what they want.