Not to waste the Tziburs time 

    with a Kapitel Tehillim

The Gaon's Minhag was not to say לדוד ה' אורי 

from R. Chodesh Ellul till Simchas Torah.

The אדרת* writes the reason for not

saying it was because of טירחא דציבורא.

Saying one extra Kapitel Tehillim takes 

up too much time of the Tzibbur.

Perhaps we can explain the Aderes /Gaon,

There is no mention in Bavli, Yerushalmi,

Rishonim, Tur, Shulchan Aruch, Kisvei Arizal etc. 

to say לדוד ה' אורי (in Ellul)

The first to mention to say it בציבור** is

the Sefer Chemdas Hayomim*** & the Sefer

Shem Tov Katan which are Sifrei  Kabbala****.

The Gaon, in general, was against adding

additional Tefillos & Pesukim. He tried to stick 

to the original Tefillos of the Anshei

Knesses Hagedola.

A Shliach Tzibbur who starts one minute

late in a Minyan of 100 people, wasted

not one minute but 100 minutes. (Tircha

of 100 people times 1 minute)

Adding a Kapitel Tehillim which takes

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.

.

 א' אלול most Shuls will be

saying 3 Kapitlach Tehillim at end

of Davening.

1)  שיר של יום

2) לדוד ה' אורי

3) ברכי נפשי 

According to the Gaon, we don't say the regular שיר של יום since the Levi'im 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 דןד 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 Nasan Ha'azasi yms.

**** I believe the first Halacha Sefer

to say it is the Mateh Efraim