Firstly it is not appropriate to answer a question with a question. Shia needs to adjust their doctrine with Quran and only after that it is appropriate to ask such a question.
Nevertheless this question only shows the misunderstanding of some brothers about the belief of the mainstream Muslims. Believing in Khulafaaye Raashedin is not a fundamental element of Islam. According to the main stream Muslims, there are only 6 Articles of Faith and 5 pillars of Islam and believing in khilaafath of Aboobakr is not part of either of them.
Any groups of people tend to elect some one as their leader. And the rational and most reasonable way to do so is by election. This is a routine social/political practice. Certainly no system of public election was established at that time and the election of Aboobakr was done through negotiation of present people. You might think that it was not a good choice or that not all qualified people were presented at the time, that's your opinion but it has nothing to do with looking for evidences in Quran about it. It's just a routine social practice that was and is and will be done in any society and no logical mind would expect a divine evidence for that.
Having said that, once the SAHABAH of the holy prophet agree on a great SAHABAH like Abu Bakr (RA) to become the Khalifah, then it is the duty of all Muslims to obey him for the sake of Islam and unity.
Anyway, Let us review the doctrine of 12er Shia first, this is the doctrine that makes 12er Shia a SECT different from the mainstream Muslims and different from other Shia sects:
"The doctrine of Imaamat: Apart from Prophets, there are another group of God appointed persons called Imams. These are people who are infallible and have access to a knowledge that is not accessible by ordinary people. The world cannot be empty of an Imam otherwise it will be destroyed. In the Islamic context, these individuals are 12 people among the descendants of the Holy Prophet who are appointed by no one but God to lead Muslims. Any one who chooses any leader other than these 12 is misguided and not a complete believer. The twelfth (last) of the above Imams is Mahdi and is alive and in occultation (now) for more than 1000 years and will come out of his occultation when God wants".
The above is the core belief of 12er Shia. They consider 5 articles of belief as fundamentals of their religion. These are: Tawheed (Oneness of God) - Nabuwwat (Prophet hood) - Ma'ad (Day of Judgement) - Adl (Justice of God) and IMAAMAT (The above doctrine).
In another way to put it they some times refer to 5 pillars of Islam to be: Prayer, Zakat, Fasting, Hajj and IMAAMAT. They further hold that the latter (Imaamat) is the most important one. Since all other four pillars of Islam ( Prayer, Zakat, Fasting, Hajj) are mentioned very clearly and repeatedly in Qur'an; IMAAMAT being fith and most important pillar of 12er Shia faith, it must also be mentioen in Qur'an very clearly and repeatedly.
The above shows the importance of Imaamat in Shia doctrine. And when I say Imaamat I don't mean Leadership. Certainly leadership is an important issue not only for Muslims but also for any group of people. Imaamat here means the above specific doctrine.
One of the main reason I try to avoid Shias that very quickly become abusive, vulgar and personal. If you do not have answer to my question then it is fine! I am OK with that but please refrain from above mentioned attributes of a classical Shia.