Tircha Dtzibura

Amongst the many Gedolim whose Minhag

was not to say לדוד ה' אורי from R. Chodesh

  Ellul till Simchas Torah, was the Vilna Gaon.

The Aderes* writes the reason for not

saying it, was because of Tircha Detzibura.

Saying one Kapitel Tehillim takes up too

much time from the Tzibbur.

Perhaps we can explain the Aderes /Gaon,

There is no mention in Bavli, Yerushalmi,

Rishonim, Tur & Shulchan Aruch etc. to say

Ledovid Hashem (in Ellul)

The first to mention to say it Betzibbur** is

the Sefer Chemdas Hayomim*** & the Sefer

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

The Gaon in general was against adding

additional Tefillos. He tried to stick to

   the original Tefillos from the Anshei

Knesses Hagedola.

A Shliach Tzibbur who starts one minute

late in a Minyan of 100 people, wasted

not one minute but100 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 Tircha Detzibura and 

didn't add it to the Tefilos during Ellul etc.

.

Sunday א' אלול most Shuls will be

saying 3 Kapitlach Tehillim at end

of Davening.

1) לדוד מזמור (שיר של יום)

2) לדוד ה' אורי

3) ברכי נפשי 

According to the Gaon all you

   would say is ברכי נפשי

    *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  contoversey who the

author was.Some don't say

Ledovid Hashem Ori

claiming the source

is Nasan Ha'azasi yms.

**** I believe the first Halacha Sefer

to say it is the Mateh Efraim