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 rakahs 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 itikaaf in al-Masjid al-Haraam or elsewhere.
The answer is: Yes, observing itikaaf in al-Masjid al-Haraam is better than observing itikaaf in other mosques. The next best is observing itikaaf 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 itikaaf 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 itikaaf 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 Kabah. 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 Kabah 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 itikaaf 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 itikaaf going out to pray Isha' and qiyaam outside of the Prophet's Mosque, his itikaaf is not valid, because he has gone out of the mosque unnecessarily.
What he should do is either observe itikaaf 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 itikaaf in the mosque where he is going to pray, so that he will attain the reward of observing itikaaf 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.
Reproduced from Islam QA