Why does the Quran not explicitly mention Hazrat Ali or the Shia Imams?

1. The Qur’an as we have it today is a compilation of speech and utterances made over 23 years in direct response to individuals, events, situations. The Quranic speech assumes the audience already knows the event/person/situation that it is talking about, so you seldom have proper names mentioned in the Quran for this reason - the Quran is addressing people who know the individuals or places or events being talked about. Tons of verses in the Quran are about Muhammad but he is mentioned by name only 4 times.
2. The Qur’anic speeches were not the only guidance that came from God. Prophet Muhammad’s own words, instructions, and commands were as religiously binding on the believers as the Qur’anic speech was. So in many cases, the Prophet’s guidance supplemented a Quranic verse to make it clear or expand on what the verse is addressing.

When 1 and 2 are kept in mind and if we examine Tafsir and Hadith that Sunni and Shia agree about, we find that numerous verses in the Qur’an refer to Imam Ali and the Prophet’s family - Hasan, Husayn, and Fatimah.

In historical context, the Qur’an does refer to Imam ‘Ali in numerous verses as attested by all Shia and numerous Sunni commenators: he is one of the Ahl al-Bayt whom God has purified from all impurity and sins (33:33) and the near-kin (qurba) of the Prophet whose love is obligatory for Muslims (42:23). Imam ‘Ali is called the Guardian (wali) of the Believers after God and the Messenger (5:55), the Witness (shahid) from the Holy Prophet who verified him (11:17), and the Guide (hadi) of the community (13:7) after the Prophet. Imam ‘Ali is part of the holy progeny of Abraham from whom God appoints the Imams (2:124) and upon whom He has bestowed His guidance, wisdom and kingdom (4:54). Imam ‘Ali is mentioned in the verse of mubahala (3:61) as the self (nafs) of the Prophet Muhammad. Verse 76:5–22 is about the Ahl al-Bayt who went hungry for days after feeding poor people every night with their own food. Verse 5:67 of the Qur’an was revealed to the Prophet Muhammad in right before the Prophet’s final announcement at Ghadeer Khum where he announced that ‘Ali is the mawla of the believers.