In point of fact...
In point of fact, Imam ‘Ali (‘a) did hold the leadership, but this did not last for a long period during Imam Hasan’s lifetime. After that, Muslims hoped that the leadership would return to the status quo when Imam Husayn (‘a) held it; therefore, they asked him to undertake the mission that resulted in the horrible tragedy of Karbala.
There is no doubt that Imam ‘Ali (‘a) raised and educated an elite group of people and built up their spiritual and organizational posture so firmly that they fought under his pennon and offered great sacrifices for the sake of Islam to substantiate the ideals and values to which the Imam (‘a) had called. Similarly, Imam Hasan (‘a) was able to preserve the same group by specifying their safety as one of the conditions of his peace truce with Mu‘awiyah.
As a result, this elite group was able to play a pioneering role during the age of Imam Husayn (‘a) when they were able to break through the stagnant states of acquiescence to the Umayyad regime under whose shadows the Muslim nation then lived.
They were also able to promulgate the objective of the movement of Imam Husayn (‘a) whose slogan was to overthrow the Umayyad ruling regime and prepare a platform on which his uprising could achieve its widest range in the milieu of the Muslim nation, leaving an endless influence on the course of Muslim history. In fact, most of these goals, including the eventual collapse of the Umayyad dynasty, were achieved.
As regards the period of the first three Imams (‘a), we can maintain that the leading and most important objective of building a virtuous community during that period was to contribute to the process of guiding others to Islam—an objective corresponding to the activities of the Holy Imams (‘a) in the field of preserving the Muslim nation.
The first procedure in this process was installing a genuinely Islamic government both externally, which was actually achieved by Imam ‘Ali and Imam Hasan (‘a), though for a short period, and theoretically.
Working towards installing an Islamic government theoretically, the Holy Imams (‘a) were able to create in the Muslim nation a trend demanding the installation of such a genuinely Islamic government and forming a pressure group that could correct the course of the Muslim governments throughout history.