It is said that "Surats Al-Baqarah and Aal-‘Imraan come as flocks of birds and intercede for their people." So based on this, is it ok for me to recite Surat Al-Baqarah on one day and Surat Aal-‘Imraan on another day and repeat this process throughout my life. Can I read Surat Al-Baqarah to obtain blessings based on the saying, "Read Surat Al-Baqarah, for taking it is blessing."

All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger. 

With regard to reciting Surat Al-Baqarah [Quran Chapter 2] on one day and Surat Aal-‘Imraan [Quran Chapter 3] on another day and repeating this continuously, then this is among the acts of righteousness for which one is expected to be rewarded. Indeed, there is much goodness in reciting Surat Al-Baqarah regularly.

The Prophet said, "Indeed, the devil flees from the house where Surat Al-Baqarah is recited." [Muslim]

There is no doubt that the one who recites it gets blessings.

For more benefit, let us shed light on the hadith that you mentioned in the question.

Abu Umaamah, may Allah be pleased with him, narrated that he heard the Prophet say, "Recite the Quran, for it will come as an intercessor for its reciters on the Day of Resurrection. Recite Surat Al-Baqarah and Aal-‘Imraan, as on the Day of Resurrection they will come like two clouds, or two black canopies with light between them, or like two flocks of birds in ranks, pleading for the one who recited them. Recite Surat Al-Baqarah, as taking it is a blessing and abandoning it is a sorrow, and the Batalah (the sorcerers or the idle and lazy people) are but helpless about it." [Muslim]

The author of Mirqaat Al-Mafaateeh commented on the hadith, saying:

"'Recite Surat Al-Baqarah', At-Tayyibi said, 'It is a specification after a specification after a generality: first, he gave an order to recite the Quran and he made intercession conditional on it; then he singled out Surat Al-Baqarah and Aal-‘Imraan and specified them for being a protection from the heat of the Day of Judgement, as they plead for their reciter; then, thirdly, he singled out Surat Al-Baqarah and linked three matters to it: Taking it – i.e. reciting it regularly, pondering about its meaning, and acting according to its teachings – is a blessing, meaning a great benefit. And abandoning it and the like of it is a sorrow, which means regret on the Day of Judgement, as it was reported that ‘the people of Paradise will not feel as sorrowful for anything as for an hour that they passed (in the worldly life) without mentioning Allah in it;’ and the (Batalah) are but helpless about it. There are two interpretations for the term Batalah: those who are idle and lazy (derived from Bataalah, which means idleness); they cannot recite it due to its length, or the sorcerers, because what they come up with is falsehood (Baatil). So he named them by what they do, which is falsehood; meaning that they will not be qualified for it, or that they will not be enabled to recite it, or that they cannot overcome it, as Allah says (what means): {But they do not harm anyone through it [magic] except by permission of Allah.} [Quran 2:102]'"

For more benefit, please refer to fatwas 344647 and 343855.

Allah knows best.