Did Imam 'Ali not use the argument of Ghadir soon after the Prophet's death??

Ali did not use the arguments during the early days of caliphate for two reasons: 

1. for the sake of Islam's well-being and survival; 

2. because of the obstinacy of his opponents. 

It had nothing to do with the implicit vs. explicit nature of appointment. The truth is that whenever the opportunity availed itself, Imam 'Ali always talked about his right to the caliphate of the Prophet.

No one can blame 'Ali for making the claim or presenting its proof at a later date. He himself said, "No person is to be blamed for delay in (securing) his own right but the blame lies on him who takes what he is not entitled to.

In the year 35 AH, while the Imam was in Kufa, he heard that some people doubted his claim of precedence over the previous three caliphs. Therefore, he came to the gathering at the mosque and implored the eyewitnesses of the event of Ghadir Khumm to verify the truth of the Prophet's declaration about his being "the mawla" (master, leader, lord) of those whose master was the Prophet himself. In most sources, we have the names of twenty four companions of the Prophet who testified to the truth of Imam 'Ali's claim. Other sources like Musnad of Ibn Hanbal and Majma'u 'z-Zawa'id of al-Haythami put that number at thirty.   [1]

One should bear in mind that this incident took place 25 years after the event of Ghadir Khumm, and during that period hundreds of eyewitnesses had died naturally or in the battles fought during that time. Add to this the fact that this incident took place in Kufa which was far from Medina, the center of the companions.

[1] This incident of Kufa has been narrated by four companions of the Prophet and fourteen of their disciples, and has been recorded in most books of history and tradition. See al-Amini, al-Ghadir, vol. 1 (Tehran: Mu'assatu 'l-Muwahidi, 1976) pp. 166-186.