This argument really just shows how devastated your are.
First regarding “Abu Bakr”
Abu Bakr was not even the real personal name of the first caliph. In your own Sunni rijal sources, the first caliph is listed as Abdullah ibn Abi Quhafa, and “Abu Bakr” is his kunya. So when you scream “Ali named his son Abu Bakr,” you haven’t even proved Imam Ali named anyone after Abdullah ibn Abi Quhafa. You’re literally confusing a kunya with a personal name just for proving your retired idea
Even worse for you, Maqatil al-Talibiyyin says about Abu Bakr ibn Ali: “lam yu‘raf ismuhu” — his actual name was not known. So what exactly are you celebrating? A kunya whose real name is disputed or unknown? That’s your proof? Pathetic.About “Abu Bakr ibn Ali”, this is even weaker than you think. Abu Bakr was not necessarily his real name; it was a kunya. Abu al-Faraj al-Isfahani says in Maqatil al-Talibiyyin: “His name was not known.” Other reports say his real name may have been Muhammad al-Asghar, or Abdullah, or Ubaydullah.
Second about “Umar”
In your own Sunni hadith material, this name is also transmitted as “Amr(عمرو)”, not “Umar(عمر)”. Ibn Abi Shaybah, in his Musannaf, narrates with the wording: “Muhammad ibn Amr(عمرو) ibn Ali, from Ali ibn Abi Talib...” So before you start screaming “Ali named his son Umar,” maybe first learn that the name itself is reported as Amr(عمرو) in your own sources. That alone destroys your fake certainty.
And even if we accept the reading “Umar” for the sake of argument, what exactly does that prove? Nothing. Umar was NOT a private trademark owned by Umar ibn al-Khattab. It was a normal Arab name. In Sunni books of Sahabah, there are at least twenty-six companions listed with the name Umar: Umar ibn Abi Salamah, Umar ibn Suraqah, Umar ibn Sa‘d, Umar ibn Sufyan, Umar ibn ‘Amr al-Laythi, Umar ibn ‘Umayr, Umar ibn ‘Awf, Umar ibn Malik, Umar al-Aslami, Umar al-Khath‘ami, Umar al-Yamani, and many others.
So your logic is embarrassingly childish. If Imam Ali having a son called Umar, then every Arab who had the name Umar must have been named after him too; even if they were born before Umar ibn Khattab!? That is not even sound like evidence; that is desperate name-game propaganda.
Third about “Uthman”
This one is even more humiliating for your argument. Uthman was a common Arab name, and Ibn Hajar in al-Isabah lists many companions named Uthman, including Uthman ibn Maz‘un, one of the early noble companions. So acting like every “Uthman” must mean Uthman ibn Affan is just childish.
Maqatil al-Talibiyyin explicitly reports that Imam Ali said about his son Uthman: “I named him after my brother Uthman ibn Maz‘un.” Not Uthman ibn Affan. Uthman ibn Maz‘un.
So again, where is your proof that Imam Ali meant Uthman ibn Affan? You have none. You just saw the same name and started building fairy tales.
And the biggest joke is this: you want to use names to erase history, while your own Sahih al-Bukhari says Hazrat Fatima became angry with Abu Bakr, stayed away from him and did not speak to him until she died. Bukhari also says Imam Ali buried Fatima at night without informing Abu Bakr, and Ali did not give bay‘ah for those months.
So stop playing cute games with names. Names do not erase Fadak. Names do not erase Hazrat Fatima’s anger. Names do not erase Imam Ali’s delayed bay‘ah. Names do not prove political approval.
Your whole argument is basically: “Someone had the same name, therefore Imam Ali loved and approved of the three caliphs.” That is kindergarten-level logic dressed up as aqidah.