Why Abu Hurayrah, Ibn Umar, Anas, Aishah, etc. have reported more traditions than Abu Bakr and Umar?
Why were most of the hadith narrated by Abu Hurayrah(RA) and Anas Malik and not by Abu Bakr(RA) and Umar(RA) despite the very were the closest companions of the Holy Prophet(SAWS)?
You have raised an important question. Abu Hurayrah, Ibn Umar, Anas, Aishah, etc. have reported more traditions than Abu Bakr and Umar; the reasons are not hard to find. Let me try to explain:
The Prophet, peace be upon him, as a divinely inspired leader and teacher, knew the talents and abilities of each one of his companions, and trained them to fill specific roles for promoting the cause of Islam. While Abu Bakr and Umar were destined for leadership, Abu Hurayrah, Anas and Aishah were to fill the roles of bearers and transmitters of prophetic legacy and traditions.
For instance, Abu Hurayrah, from the beginning of his association with the Prophet showed great zeal to retain whatever he learned from the Prophet; and on realizing this, the Prophet prayed to Allah to bless his memory. Thanks to the Prophet's prayer, he was gifted with remarkable power of retention.
Ibn Umar, from his early years, possessed an unbounded passion to observe and emulate the mannerisms and examples of the Prophet. He was reputed for his piety and scrupulousness in following the Prophet's sunnah--a fact unanimously accepted by the companions.
Anas accompanied the Prophet ten years; he was ten when he came to the service of the Prophet as his mother had wanted him to help the Prophet in his errands and thus be able to observe and learn from him.
Aishah, the beloved wife of the Prophet, was the closest to the Prophet, second to none. The Prophet also trained her closely to fill the role of a teacher, as can be seen from a close study of the sources. Although the general assumption is that Abu Hurayrah was the most prolific transmitter of traditions, some scholars, after critical study of the sources, have concluded that Aishah has surpassed all others, including Abu Hurayrah, Ibn Umar and Anas.
Coming to Abu Bakr and Umar, they were trained by the Prophet, peace be upon him, as leaders--a role they filled very well. Every student of Islamic jurisprudence also recognizes that their legal precedents formed an organic expansion of the Prophetic legacy.
May Allah open our hearts to recognize the divine wisdom in the preservation of the sunnah as part of His promise to preserve the Quranic revelation.