The Gaon's Minhag was not to add לדוד ה' אורי from R. Chodesh Ellul till Simchas Torah Davening.
The אדרת* writes the reason for not saying it was because of טירחא דציבורא.
up too much time of the Tzibbur.
Perhaps we can explain the Aderes /Gaon,
There is no mention in בבלי, ירושלמי, ראשונים, טור, שו"ע, כתבי אריז"ל etc. to say לדוד ה' אורי (in Ellul)
The first to mention to say it בציבור** is the Sefer חמדת הימים*** & the Sefer שם טוב קטן which are ספרי קבלה****.
The Gaon, in general, was against adding additional Tefillos & Pesukim. He tried to stick to the original Tefillos of the Anshei Knesses Hagedola.
to say only one minute can at times
be considered taking 100 minutes.
This might explain why the Gaon considered it טירחא דציבורא and didn't add it to the Tefilos during Ellul etc.
most Shuls will be saying 3 קפיטלך תהילים at end of Davening.
According to the Gaon, we don't say the regular שיר של יום since the לויים didn't say it on Rosh Chodesh & Yom Tov. All you would say is ברכי נפשי which was the שיר של יום of Rosh Chodesh.
*
הרב אליהו דוד רבינוביץ תאומים
(he was a twin)
** The Sefer Shem Tov Katan printed before the Chemdas Hayomim mentions only to say it ביחיד.
*** major controversy who the author was. Some don't say Ledovid Hashem Ori (Ellul) claiming the source is נתן העזתי ימ"ש
**** I believe the first Halacha Sefer to say it is the מטה אפרים
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