You need to take a look at your question. The verse proves three things; 1 It proves that there is such thing as Imamah. 2, Allah chooses and makes Imams. 3, There will be Imams in the offsprings of Abraham. Also note how the verses begins "And remember oh Muhammad s.a.w ". What or why is Allah reminding Muhammad?
1. So the word "Imam" simply exists in the Quran, and?
2. And?
3. And?
Where is 12 infallible Imams proven in any of this? You asked "What or why is Allah reminding Muhammad?". So Allah reminded the prophet of Ibrahim A.S's trials therefor 12 infallible Imams? That doesn't really answer me.
I've discussed the possibility of 2:124 referring to prophets due to the verse I gave and you didn't address it yet. You stated "There will be Imams in the offsprings of Abraham", but how does this prove 12 infallible Imams? If we were to go by that, the verse makes a condition for someone to become an Imam which is righteousness. So literally anyone who's righteous from prophet Ibrahim A.S's offspring all the way to the prophet Muhammad S.A.W, and from the prophet S.A.W up until the day of judgement, could be an Imam. It can be in the thousands, it could be anyone, because the verse doesn't in any way provide any specifications other than Imam has to be righteous and from Ibrahim A.S's progeny. Of course that is if we assume the verse isn't referring to
prophets from Ibrahim's progeny at all. But again:
"And We have already sent Noah and Abraham and placed in their descendants prophethood and scripture; and among them is he who is guided, but many of them are defiantly disobedient." - Quran 57:26
"And We gave him Isaac and Jacob in addition, and all [of them] We made righteous.And We made them leaders guiding by Our command. And We inspired to them the doing of good deeds, establishment of prayer, and giving of zakah; and they were worshippers of Us." - Quran 21:72-73
Due to those verses, it's much easier to believe that the verse in Surah al Baqara is referring to the prophets from his progeny. Not your 12 infallible Imams. Also, where does the idea that Imams automatically have the right to political power over anyone else come from? Allah says:
"And We made from among them leaders guiding by Our command when they were patient and [when] they were certain of Our signs" - Quran 32:24
Similar to the previous verse from Surah al Anbiyaa', this suggests that Imams are tasked with calling people to the truth and not for the sake of political power. So what Quranic verse is there to prove that Imams automatically have the right to political power over anyone else and that anyone who opposes them (politically) is a hypocrite that's opposing Allah?
Now take a look at your question, why do you need verses in black and white about something (12 Imams) after the Quran was revealed. Isn't logic and reason plus Muhammad's saw saying enough for you to believe?
"The month of Ramadhan [is that] in which was revealed the Qur'an, a guidance for the people and clear proofs of guidance and criterion...." Quran 2:185
Can you prove everything from the Quran. Is it necessary that everything has to been proven from the book otherwise it ain't acceptable. We believe Muhammad s.a.w left the book and his progeny and we follow that. You grasp and take what a handful of companions were upto. It's just as simple as that.
I believe that everything that is relevant to one's faith or makes one a Muslim is well established in the Quran. Such as the 5 pillars of Islam, 6 pillars of faith, obeying Allah and His Messenger, etc. If obeying the Imams was so important to one's faith then why wouldn't it be well established in the Quran like the rest of the important stuff I've just mentioned? Should we resort to shady (most likely fabricated) Shia narrations or faulty verdicts based on forced interpretations of ambiguous verses that are influenced by theological biases?
The main reason I'm sticking with the Quran is because it is an indisputable proof upon all of us here. Or do you consider Bukhari, Muslim and others to be proof upon you?