I have read the following text in Al-Bidayah wan-Nihayah by Ibn Katheer:
On the Day of Saqeefah, The companions, may Allah be pleased with them, unanimously decided that the Muslim ruler must be from the tribe of Quraysh. Abu Bakr, the great Companion, quoted a Prophetic statement to support such a decision. At that time, the Ansar (the people of Madinah) requested that they would select a person from among themselves to rule hand in hand with the ruler from Quraysh. However, Abu Bakr did not approve their request.
My question is: what is the prophetic text that Abu Bakr used to support his stance? Also, does this text not contradict the other prophetic text that states, “No Arab is superior to a non-Arab except by piety?” How is it that the Muslim ruler must be from Quraysh? Why should a Qurashi (except for the Prophet’s relatives) be superior over other Muslims? Are the superiority and selection of Qurashi people for leadership mandated by Allah Himself, just as the Jews would claim they were Allah’s chosen people, even though they later renounced their religion and were negligent concerning Allah’s rights?

All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.

The event that occurred at the Saqeefah of Banu Sa‘idah following the death of the Prophet, sallallaahu ‘alayhi wa sallam, is well-known in Muslim history. In brief, the companions who attended that event ended up choosing Abu Bakr to be the Caliph of the Prophet, sallallaahu ‘alayhi wa sallam. Both Al-Muhajiroon (the emigrants from Makkah) and Al-Ansar (the citizens of Madinah) pledged allegiance to Abu Bakr as the new Muslim ruler. Their choice was founded on textual proofs, rather than personal views. Here is the most important text that the companions referred to in this regard:

On his deathbed, the Prophet, sallallaahu ‘alayhi wa sallam, asked Abu Bakr to lead the Muslims in prayer on his behalf. The companions considered the Prophet’s command for Abu Bakr to succeed him in prayer as an indication that Abu Bakr should succeed him in the leadership of the Muslim community. Therefore, when the companions chose Abu Bakr for the caliphate, they said: “The Prophet, sallallaahu ‘alayhi wa sallam, appointed him to lead us in the matters of religion, so do we not accept him to lead us in the matters of this life?” This view was reported by Ibn Abdul-Barr in At-Tamheed on the authority of Ali Ibn Abi Talib. It has been reported by other scholars as well. This type of proof is called ‘analogy of priority’ (where the effective cause is more evident in the new case than in the original case) by the majority of the scholars of Islamic fundamentals, and according to Hanafi scholars, it is called ‘the text’s indication’. The companions supported their position by way of analogical reasoning where they made the case of Muslim rule analogical to the case of prayer leadership insofar as Abu Bakr’s selection in both cases was concerned.

Abu Bakr’s deduction that the Muslim ruler must be from Quraysh is established by the consensus of Muslim scholars. It is also substantiated by several prophetic statements that have been narrated through authentic chains of transmission.

For example, the Prophet, sallallaahu ‘alayhi wa sallam, said, “The (Muslim) rulers are to be from Quraysh.” [Ahmad; Al-Arna’oot: authentic] The origin of this Hadeeth was also related by Al-Bukhari and Muslim, but with a different wording. Abu Bakr used this prophetic statement to prove his position, as was reported by Ibn Al-‘Arabi in Al-‘Awasim min Al-Qawasim. In the same respect, Wathilah ibn Al-Asqa‘ said that he heard the Prophet, sallallaahu ‘alayhi wa sallam, say, “Allah selected Kinanah from the progeny of Isma‘eel (Ishmael) ; Quraysh from Kinanah; Banu Hashim from Quraysh; and myself from Banu Hashim.” [Muslim]

The consensus of the Muslim scholars, as far as this issue is concerned, is based on the Prophetic statement, narrated by Abu Hurayrah, in which the Prophet, sallallaahu ‘alayhi wa sallam, said, “People should follow Quraysh in the matter of rule; the Muslims and non-Muslims among them.” [Al-Bukhari and Muslim] In his commentary on Saheeh Muslim, An-Nawawi said:

This statement and others imply that the caliphate is restricted to the tribe of Quraysh. None is allowed to assume the post of Caliph unless he is from Quraysh. The companions were unanimous on this matter, and those who come after them should follow the same rule.

Privileging Quraysh with the caliphate does not contradict Allah’s saying about the believers (what means): {Indeed, the most noble of you in the sight of Allah is the most righteous of you.} [Quran 49: 13] The verse makes all humans equal in humanity; however, everyone is different in how close to Allah he may be according to how righteous he is. Nevertheless, it is Allah who has chosen Quraysh for the caliphate, as stated in the previous statements of the Prophet, sallallaahu ‘alayhi wa sallam. Such choice goes in line with Allah’s saying (what means): {And your Lord creates what He wills and chooses...} [Quran 28: 68]

Yet, some Muslim scholars explained that Quraysh was chosen for the Muslim caliphate because they had a strong position among the Arabs. Quraysh was located in the center of Arabia, and it (Makkah) was considered the headquarter for entire Arabia. In his Muqaddimah, Ibn Khaldoon provided a detailed account on this issue. Finally, we must assert that Islam does not differentiate between people based on race or ethnicity. Rather, Islam basically came to eradicate racism and fanaticism in all their aspects. Bilal was an Abyssinian, Suhayb was a Roman, and Salman was a Persian. All of these companions were, however, bound by the rope of Muslim brotherhood.

Allah knows best.