He does not pray in the Prophet’s Mosque so that he can pray behind an imam who he thinks helps him to have better khushoo‘ (focus) in his prayer

Prayers offered in the Prophet’s Mosque bring a greatly
multiplied reward. 

Al-Bukhaari (1190) and Muslim (1394) narrated from Abu
Hurayrah (may Allah be pleased with him) that the Prophet (blessings and
peace of Allah be upon him) said: “One prayer in this mosque of mine is
better than a thousand prayers anywhere else, except al-Masjid al-Haraam.” 

This hadeeth includes both obligatory prayers and naafil
prayers that it is mustahabb (recommended) to offer in congregation in the
mosques, such as Taraweeh. 

The scholars of the Standing Committee for Issuing Fatwas
said: 

It is mustahabb to offer naafil prayers at home, whether they
are regular Sunnah prayers (ar-rawaatib) or otherwise, except those
which Allah has prescribed should be offered in the mosque, such as the two
rak‘ahs offered when entering the mosque (tahiyyat al-masjid or
greeting the mosque); those which Allah has prescribed should be offered in
congregation, such as Taraweeh and the eclipse prayer, which should be
offered in the mosque; and Eid prayers and prayers for rain (istisqa’),
which should be offered in the musalla (prayer place).

End quote from Fataawa al-Lajnah ad-Daa’imah, 7/239 

With regard to your companion not praying Taraweeh in the
Prophet’s Mosque for the reasons you mentioned, undoubtedly khushoo‘ (proper
focus) in prayer and presence of mind in prayer is something to be sought,
and a beautiful voice in recitation of the Qur’an helps with that and has an
impact on the listener. If your companion was going to pray in another
mosque and that would help him to have better focus, then there is no blame
on him. He may have forsaken the virtue of the place and been keen to attain
a virtue that has to do with the prayer itself, namely khushoo‘. If there is
a conflict between two virtues – as is the case with your companion – then
precedence should be given to the virtue that has to do with the act of
worship itself. 

Shaykh Ibn Jibreen said: 

A beautiful voice and good recitation have an impact on the
heart and help to bring about presence of mind and physical stillness, and
enable one to be affected by the words of Allah, may He be exalted, and to
enjoy listening to them, which may be a means of understanding them,
comprehending their meaning and appreciating their miraculous nature,
eloquence and style. All of that is in turn leads acting upon them and
accepting their guidance. So the one who looks for a reciter with a
beautiful voice who recites the Qur’an well, has memorised it, whose
recitation helps him to focus and is unhurried in his prayer is not to be
criticized. One should look for such an imam to pray behind, even if the
place is far away, and this imam is to be preferred over one who does not
recite well, or makes many mistakes, or does not have a pleasant voice, or
does not make his voice beautiful when reciting Qur’an, or who recites too
quickly, or is not unhurried in his prayer, or does not sound focused in his
recitation, even if his mosque is nearby. End quote. 

Fataawa ash-Shaykh Ibn Jibreen,
24/28 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: 

The questioner asked whether it is better to observe i‘tikaaf
in al-Masjid al-Haraam or elsewhere. 

The answer is: Yes, observing i‘tikaaf in al-Masjid al-Haraam
is better than observing i‘tikaaf in other mosques. The next best is
observing i‘tikaaf in the Prophet’s Mosque, then doing so in al-Masjid
al-Aqsa, then other mosques, the next best then the next best. 

But here there is an issue that we should highlight, which is
that paying attention to the act of worship itself is more important than
paying attention to its time and place. In other words, the virtues
pertaining to the act of worship itself are more important than the virtues
pertaining to its time or place. That is, if a person’s i‘tikaaf in a mosque
other than the three mosques will be more perfect, with a stronger focus on
humility before Allah, may He be glorified and exalted, in his worship, then
observing i‘tikaaf in those mosques is preferable, because this advantage
has to do with the act of worship itself. 

The scholars think that being able to walk rapidly (raml)
in tawaaf al-qudoom (tawaaf of arrival, in the first three circuits) is more
important than getting close to the Ka‘bah. They explained that by noting
that walking at a rapid pace is a virtue that has to do with the act of
worship itself, whereas getting close to the Ka‘bah is a virtue that has to
do with the place; paying attention to matters that have to do with the act
of worship itself is more important than that which has to do with the
place. This is a point which people, especially seekers of knowledge, should
pay attention to, namely focusing on the virtue of the act of worship itself
more than focusing on the place or time. End quote. 

Fataawa Noor ‘ala ad-Darb,
205/4-5 

The shaykh also said: 

Undoubtedly i‘tikaaf observed in one of the three mosques to
which journeys may be made is better, and there is no difference of
scholarly opinion on this point, except if observing i ‘tikaaf in his own
mosque will lead to greater khushoo‘ and focus of mind, and he will be safe
from noise and disturbance, and seeing inappropriate scenes. In that case,
we say that his own mosque is preferable. End quote. 

Sharh al-Kaafi, 4/159 

What appears to us to be the case is that there is nothing
wrong with what your companion did, even though the imams of the Haramayn
are carefully selected from among those who recite well and have pleasant
voices. 

With regard to one of those who are observing i‘tikaaf going
out to pray ‘Isha’ and qiyaam outside of the Prophet’s Mosque, his i‘tikaaf
is not valid, because he has gone out of the mosque unnecessarily. 

What he should do is either observe i‘tikaaf for all of the
last ten days in the Prophet’s Mosque, and not go out except in the case of
necessity, which is best; 

Or he should observe i‘tikaaf in the mosque where he is going
to pray, so that he will attain the reward of observing i‘tikaaf in the last
ten days, and thus he will have followed the example of the Prophet
(blessings and peace of Allah be upon him). 

And Allah knows best.