I am a Shia and have this doubt about Ahlul-Sunnah, and no Sunni Mufti is agreeing to answer a few questions I have, such as: amongst the companions Abu Bakr is viewed as the most superior on account of his closeness to the Prophet, sallallaahu alayhi wa sallam. If this is indeed the case then why did the Prophet not select him to be his brother when he divided the companions in to pairs on the Day of Brotherhood? Rather, the Prophet chose Ali, may Allaah be pleased with him, saying "You are my brother in this world and the next", so on what basis is Abu Bakr closer?
See the History of the Caliphs who took the right way, by Jalaladeen Suyuti, English translation by Abdassamad Clarke p177, (Taha publishers).

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

The excellence of Abu Bakr over all other Companions is something on which the Muhaajiroon (Emigrants) and the Ansaar (Helpers) agreed in a consensus. Ibn 'Umar narrated, “During the reign of the Prophet, sallallaahu ‘alayhi wa sallam, we did not equate anyone to Abu Bakr, then to 'Umar then to 'Uthmaan and then we did not distinguish between the remaining Companions [we considered the rest as equal in rank]." [Al-Bukhari]

Moreover, Al-Bayhaqi reported that Ash-Shaafi'i said, “None of the Companions or their righteous followers differed with regard to the preference of Abu Bakr and 'Umar over all other Companions ; nonetheless some of them differed in relation to 'Ali and 'Uthmaan as to who was better than the other.”

Indeed there are many texts and evidences that prove the excellence and preference of Abu Bakr over all other Companions in excellence and as a caliph; we are going to mention some of them.

The Prophet, sallallaahu ‘alayhi wa sallam, said: “If I were to take an intimate friend, I would have taken Abu Bakr as one, but he is my brother and my companion.” [Al-Bukhari and Muslim] It is also confirmed that 'Amr ibn Al-'Aas asked the Prophet, sallallaahu ‘alayhi wa sallam, “Who is the dearest person to you?'' He, sallallaahu ‘alayhi wa sallam, replied: "‘Aa'ishah." ‘Amr asked, "And among men?” He, sallallaahu ‘alayhi wa sallam, said: "Her father.” [Al-Bukhari and Muslim]

Furthermore, 'Aa'ishah narrated, “When the Prophet, sallallaahu ‘alayhi wa sallam, was ill during his death sickness, when it was time for the prayer and the call to the prayer was made, He, sallallaahu ‘alayhi wa sallam, said: “Order Abu Bakr to lead the Muslims in prayers.” [Al-Bukhari and Muslim]

As for the narration that you mentioned in the question, it is reported by At-Tirmithi that the Prophet, sallallaahu ‘alayhi wa sallam, said to 'Ali : “You are my brother in this world and in the Hereafter.” Some scholars classified the narrators of this narration to be weak. Even if we presume that this narration is authentic, the most that could be said about it is that it proves the excellence of 'Ali ; this is a matter that no one doubts. Indeed, there are many texts about his excellence and virtues.

However, there is nothing in the above narration, that suggests that ‘Ali is better than Abu Bakr and this is not a defect or imperfection in Abu Bakr . If this were the case, it would be said, why didn't 'Ali accompany the Prophet, sallallaahu ‘alayhi wa sallam, when he, sallallaahu ‘alayhi wa sallam, migrated to Madeenah, and why did the Prophet, sallallaahu ‘alayhi wa sallam, not say, “I would have taken 'Ali as a close friend Instead of Abu Bakr”, and so forth.

Therefore, the excellence and preference of Abu Bakr and 'Umar, over other Companions is evident, and Abu Bakr is the best of all other Companions without exception. This is indeed what 'Ali himself believed in and said. 'Abdullaah ibn Salamah narrated, “I heard 'Ali saying, ‘The best of the people after the Prophet, sallallaahu ‘alayhi wa sallam, is Abu Bakr and the best of the people after Abu Bakr is 'Umar ." [Ibn Maajah]

Allaah Knows best.