Officially, Abu Bakr is the best of this Ummah, after its Prophet, sallallahu ‘alaihi wa alihi, according to the Ahl al-Sunnah. ‘Allamah al-Albani (d. 1420 H) for instance submits:

I say: the fact that Abu Bakr, may Allah be pleased with him, was the most beloved of mankind to him (i.e. the Prophet), peace be upon him, is consistent with the fact that he was the best of the rightly guided khalifahs in the view of the Ahl al-Sunnah.1
However, this belief directly contradicts their “authentic” hadith. Imam Ahmad (d. 241 H) records:

‘Abd Allah (b. Ahmad) – my father (Ahmad b. Hanbal) – Yahya b. Hamad – ‘Abd al-‘Aziz b. al-Mukhtar – Khalid al-Khadha’ – Abu ‘Uthman – ‘Amr b. al-‘As:
The Messenger of Allah, peace be upon him, appointed me as commander of the army of Dhat Salasil. So, I got to him, and said, “O Messenger of Allah, which of mankind is the most beloved to you?” He replied, “‘Aishah.” I said, “Who among the men?” He replied, “Her father.” I asked, “Then who?” He replied, “’Umar”.2
Al-Arnauṭ comments:

Its chain is sahih upon the standard of the two Shaykhs.3
In other words, Umm al-Muminin ‘Aishah is the overall best of this Ummah, above Abu Bakr, ‘Umar and ‘Uthman, according to the Sunni-only report. It is, however, at this point that things get really messy! Allah has stated concerning two of the wives of His Prophet:

And when the Prophet disclosed a matter in confidence to one of his wives, so when she told it, and Allah made it known to him, he informed part thereof and left a part. Then when he told her thereof, she said, “Who told you this?” He said, “The All-Knower, the All-Aware has told me.” If you two repent to Allah, for your hearts have deviated. But if you both help each other against him, then Allah is His Helper (against you both), and Jibril, and the righteous believers, and furthermore, the angels are his helpers. It may be if he divorced you that his Lord will give him instead of you, wives better than you: Muslims, believers, obedient, repentant, devoted, fasting – whether previously married or virgins.4
Imam Ibn al-Jawzi (d. 597 H) has this exegesis:

Then He (Allah) addresses ‘Aishah and Hafsah, saying: {If you both repent to Allah}, meaning from helping each other against the Messenger of Allah to hurt him. {For your hearts have deviated}Ibn ‘Abbas said: “They (the hearts) deviated (zaghat) and committed a sin.” Al-Zajaj said, “They (the hearts) deviated, and deviated from the Truth.” Mujahid said, “We used consider His Words, the Almighty {for your hearts have deviated} has something easy until we found it in the recitation of Ibn Mas’ud as: {for your hearts have deviated (zaghat)}”.5
Imam Muslim (d. 261 H) also records:

Harun b. Sa’id al-Ayli – ‘Abd Allah b. Wahb – Sulayman b. Bilal – Yahya – ‘Ubayd b. Hunayn – ‘Abd Allah b. ‘Abbas:
I hesitated for a (whole) year, and I had intended to ask ‘Umar b. al-Khaṭṭab concerning a verse. But I could not ask him out of fear of him, until he went out for Hajj and I accompanied him. During his return, while we were still on the way, he stepped aside towards an Arak tree to ease himself. So, I waited for him until he finished. I then walked along with him, and said, “O Amir al-Muminin! Who were the two women who helped each other against the Messenger of Allah, peace be upon him, among his wives?” He replied, “They were Hafsah and ‘Aishah.”6

There are three things here:
1. Both ‘Aishah, and especially Hafsah, betrayed the confidence of the Messenger of Allah.
3. Both of them literally helped each other against him in order to hurt him.
5. The hearts of both of them had deviated from the Truth. This is very obvious, anyway. No one with a clean heart would ever help another against the Messenger in any circumstance.
Interestingly, the above verses were the last updates by Allah on the hearts of both women. Nothing else was revealed thereafter by Him to discharge them, or to indicate their repentance. It is a matter of great interest then that the deviation of their hearts means they both have little or no hope of salvation in the Hereafter:

The Day whereon neither wealth nor sons will avail, except him who brings to Allah a clean heart.7
The perturbing question here is: how is ‘Aishah the best of this Ummah, after its Prophet, despite that she was a deviant in the Sight of Allah? Are our Sunni brothers telling us that Abu Bakr, ‘Umar and ‘Uthman were worse than deviants? Besides, Allah mentions the existence of women who would be better wives to His Prophet; if case he divorced ‘Aishah and Hafsah. Does this fact alone not debunk the Sunni ahadith on the superiority of Umm al-Muminin ‘Aishah as mere sectarian polemical artwork?

References
1. Muhammad Nasir al-Din b. al-Hajj Nuh al-Albani, Silsilah al-Ahadith al-Ḍa’ifah wa al-Mawdhu’ah wa Atharihah al-Sayyiah fi al-Ummah (Riyadh: Dar al-Ma’arif; 1st edition, 1412 H), vol. 3, p. 255, # 1124
2. Abu ‘Abd Allah Ahmad b. Hanbal al-Shaybani, Musnad (Cairo: Muasassat Qurtubah) [annotator: Shu’ayb al-Arnaut], vol. 4, p. 203, # 17844
3. Ibid
4. Qur’an 66:3-5
5. Abu al-Faraj Jamal al-Din ‘Abd al-Rahman b. ‘Ali b. Muhammad al-Jawzi al-Qurshi al-Baghdadi, Zad al-Masir fi ‘Ilm al-Tafsir (Beirut: Dar al-Fikr; 1st edition, 1407 H) [annotator: Dr. Muhammad b. ‘Abd al-Rahman ‘Abd Allah], vol. 8, p. 52
6. Abu al-Husayn Muslim b. al-Hajjaj al-Qushayri al-Naysaburi, Ṣahih Muslim (Beirut: Dar Ihya al-Turath al-‘Arabi) [annotator: Muhammad Fuad ‘Abd al-Baqi], vol. 2, p. 1105, # 1479 (31)
7. Qur’an 26:88-89