The Prophet saw departed without appointing a successor, this is a fact backed by Mutawatir narratives. The Prophet saw never said its not allowed to appoint, but he emphasyzed on the importance of consultation. Abu Bakr had the choice of either appointing or leaving the matter to them. The believers asked him to make a decision and they told him they'd accept whomever he chose, that's why there were no disputes when he picked Umar. Umar was unsure whether he should choose or not, after consultation he decided to nominate six men. Uthman was killed before anything could happen. Ali accepted to be a leader after they insisted as long as he got general acceptance in the Masjid like the three before him did. Ali decided to leave the people without appointing a successor, he refused to appoint anybody even after they insisted.
Nahj ul Balaghah is a weak chainless story book. Ali thought Abu Bakr was worthy in more than one authentic narration.
The part of Abu Bakr sermon is a part you misunderstood since you're a weirdo.
You also misunderstood Umars words and Umar also said "if it weren't for Muadh then Umar would have perished."
As opposed to what you said Ali was not chosen by everyone. The people of Syria all refused him. A lot of Sahabah did not select him or offer Bayah or obedience. Zubayr and Talhah were forced to give Bayah. All in all, Abu Bakr, Umar and Uthman had a lot more acceptance.
As for Abu Bakr, some of the Ansar refused to elect anyone other than their own leader Saad. The Mouhajiroun convinced the Ansar to select Abu Bakr and so they all chose him. Only banu Hashim refused to select Abu Bakr at first but then they corrected their mistake and returned to goodness. Umar was accepted by all unanimously and so was Uthman, everyone gave them Bayah.
As for the prophet saw appointing deputies during his life, it's different than appointing a successor after his death. Two different scenarios.