The big issue on this case is Umar (ra) made it binding for the next ruler to adopt the ijithad of himself and Abi Bakr (ra). Due to this Imam Ali (as) again was sidelined.
Brother, did Omar made it binding or was it Abdul-Rahman ibn 'Awf?! Most Shias (Arabs) online use this narrations from Al-Ya3qubi (A Shia whom the Rafidah try to sell as a Sunni. The following quote is from a notorious Rafidi website al-hadi us ... full of misquotes and fabricated narrations):
Abdul Rahman ibn A'awf asked Ali: "If I have give you this caliphate, will you follow the Book of Allah, the Sunnah of his prophet and the ways of Abu Bakr and Umar?" Ali responded that he will only follow the Quran and the Sunnah of the ptophet. The same question was asked to Uthman and he responded that he will follow the path of the Quran, the prophetic Sunnah and the ways of Abu Bakr and Umar. Abdul Rahman asked them again the same question and he received the same answers. So Abdul Rahman picked Uthman as the next caliph after Umar because he agreed to follow the ways of Abu Bakr and Umar while Ali did not.
إن عبد الرحمن بن عوف قد خلا بعلي بن أبي طالب فقال: لنا الله عليك إن وليت هذا الأمر " الخلافة " أن تسير فينا بكتاب الله وسنة نبيه وسيرة أبي بكر وعمر. فقال علي أسير فيكم بكتاب الله وسنة نبيه ثم خلا بعثمان فقال له: لنا الله عليك إن وليت هذا الأمر أن تسير فينا بكتاب الله وسنة رسوله وسيرة أبي بكر وعمر، فقال عثمان: لكم أن أسير فيكم بكتاب الله وسنة نبيه وسيرة أبي بكر وعمر... وأعاد القول على الاثنين مرة ثانية فأجابا بمثل جوابهما أول مرة، فصفق عبد الرحمن على يد عثمان كناية على اختياره للخلافة لأن عثمان قد التزم بأن يعمل بكتاب الله وسنة رسول الله وسيرة أبي بكر وعمر، أما علي فقد التزم فقط بالعمل بكتاب الله وسنة رسول ولم يلتزم بالعمل بسيرة الشيخين أبي بكر وعمر لذلك صرفت عنه الخلافة.
The narration above can be found in other sources as well, even Shia websites don't claim what you claimed, look here (taken from the Rafidi website al-islam org):
Uthman, the Third Khalifa of the Muslims
....
Other members agreed but Ali hesitated to give Abdur Rahman any extra powers. When others insisted that he too should agree, he said to him:
“I shall acknowledge you as chairman of the electoral committee if you give a pledge that you will not be a slave to your own lusts, and that your decision will be taken only to win the pleasure of God and His Messenger.”
Abdur Rahman readily gave his pledge for doing so, and thus became the chairman of the committee. ”
In a very detailed Hadith in Al-Bukhari about the Bay3ah after Omar there is not even such a thing about Ali not wanting to accept the Bay3a because of the condition (of following the sirah/ijtihadat of the Shaykhayn). And as I said, even the Rafidah don't claim that it was Omar who made the suggestion, rather it was one of the heads of the Muhajirs, Abdul-Rahman Ibn Awf, and here the proof (note how Ali is not rejecting anything at all):
أحمد بن حنبل - مسند العشرة .. - مسند عثمان.. - رقم الحديث : ( 526 )
- حدثنا : عبد الله ، حدثني : سفيان بن وكيع ، حدثني : قبيصة ، عن أبي بكر بن عياش ، عن عاصم ، عن أبي وائل قال : قلت لعبد الرحمن بن عوف : كيف بايعتم عثمان وتركتم علياًً (ر) قال : ما ذنبي قد بدأت بعلي فقلت : أبايعك على كتاب الله وسنة رسوله وسيرة أبي بكر وعمر (ر) قال : فقال فيما إستطعت ، قال : ثم عرضتها على عثمان (ر) فقبلها.
Abu W'ail says: I asked Abdur Rahman Auf: "Why did you give your oath to Othman and ignored Ali", he replied: "I am not to be blamed, I offered my oath to Ali on the condition that he will follow book of God, the Sunnah of the Prophet, and the way of Abu bakr and Omar, and he replied "
to the best of my abilities", I offered the same to Othman and he agreed.
Al-Tabari reported the story in his Tarikh:
Abdul Rahman ibn A'awf said to Ali: "do you promise to act in accordance with the book of Allah, the Sunnah of His Messenger and the Sirah of the two previous caliphs?" Ali replies to this question by saying "
To the extent that my knowledge and strength allows." Uthman was asked the same question and he replied: "Yes."
فقالَ عبد الرحمن: إنّي قد نظرتُ وشاورتُ، فلا تجعلُنَّ أيُّها الرهطُ على أنفسكم سبيلاً، ودعا علياً فقالَ:
ـ عليكَ عهدُ اللّهِ وميثاقُه لتعملنَّ بكتاب اللّهِ وسنةِ رسوله وسيرةِ الخليفتين من بعده، قالَ:
ـ أرجو أنْ أفعلَ وأعملَ بمبلغ علمي وطاقتي، ودعا عثمانَ، فقالَ له مثلَ ما قالَ لعلي، قالَ:
ـ نعم، فبايعه، فقالَ عليٌّ:
حبوتَه حبوَ دهر، ليس هذا أولَّ يومٍ تظاهرتُم فيه علينا، فصبرٌ جميلٌ، واللّهُ المستعانُ على ما تصفون، واللّهِ ما ولَّيتَ عثمانَ إلّا ليردَّ الامرَ اليكَ
Ali's praise for Abu Bakr and Omar is mass narrated (like on the pulpit of Kufa), so every sincere researcher will understand nothing from the above except that Ali was being humble or let's say less sure if he can achieve the great achievements of Abu Bakr and Omar. If the whole thing was based on falsehood than Ali would have never participated in the first place, as I've explained above.
Why did Omar pick Ali amongst the six people of committee ? He could have just excluded him altogether, Shias even believe that Omar and Abu Bakr tried to kill the Prophet AND Ali (acc. to one of their narrations Abu Bakr and Omar send Khaled to kill Ali at Fajr prayer!). So why on earth does Omar even include Ali in the committee?! The answer is obvious, neither Omar nor Ali nor anyone else heard of the exclusive right of Ali (let alone his sons!) to rule over the Ummah. Othman even voted for Ali!
SELECTION OF UTHMAN
Umar asked that, after his death, the committee reach a final decision within three days, and the next Caliph should take the oath of office on the fourth day. If Talhah joined the committee within this period, he was to take part in the deliberations, but if he did not return to Medina within this period, the other members of the committee could proceed with the decision. Abdur Rahman bin Awf withdrew his eligibility to be appointed as Caliph in order to act as a moderator and began his task by interviewing every member of the committee separately. He asked them for whom they would cast their vote. When Ali was asked, he didn't reply. When Uthman was asked, he voted for Ali, Zubayr said for Ali or Uthman. and Saad said for Uthman.
REFRENCE: A Restatement of the History of Islam and Muslims on Al-Islam.org referencing Al-Fitna Al-Kubra (The Great Upheaval), published by Dar-ul-Ma'arif, Cairo, 1959, p. 47
Omar included Ali because Omar LOVED Ali more than his own son this is why he always treated Ali well and Ali accepted the gifts such as property and land:
http://twelvershia.net/2013/06/02/umar-ra-returned-to-ali-ra-the-lands-and-gave-him-new-ones/and Ali participated because he loved Abu Bakr and Omar and didn't believe that the Shura is taghuti nor did he believe that his political leadership was usurped. If Ali claimed that elections is invalid then participating in one to elect himself would make him a hypocrite. The Shias own interpretation of history on this matter is making Ali a hypocrite, but they aren't able to see this because of they bias. All they do is come with 'excuses' for these inconsistencies.