What would be the purpose of having extra-Qur'anic information, such as Mushaf Fatimah and al-Jafr, if it is only accessible to the Imams? Wouldn't that information regarding Fiqh and Sharia be beneficial to the people as well?
One has to assume that if the possessors of these books chose not to share them, then it is not necessary for us to know the contents. Mushaf Fatimah and al-Jafr, according to hadith, contain more than fiqh and shari'ah; for instance, according to hadith, they contain detailed future prophecies, and perhaps this information would not be beneficial for us. According to hadith, Mushaf Fatimah was narrated to her as a form of consolation due to her grief for her father, so from that angle, there may have been a personal purpose.
There are a number of sacred items mentioned in hadith as being passed on from the prophets to the Imams as a sign of spiritual authority and these can be looked at in this light.
There is no indication that the Imams required these materials to learn, but that does not mean that there is any problem in their having them or consulting them. Perhaps it was reassuring to some people around them, who did not believe in the imamate, that they were referring to texts received from their forebears.
We live in an era where the value of things is measured very materially, especially in the era of the Internet, when it is assumed that anything worthwhile should be shared publicly. However, this brings up the question of, do things have value if people do not see them? (For instance, a tree in a forest.) Is it possible for something to have purpose if it is seen by only one or a handful of people? It is reasonable to argue that there is a spiritual value to the existence of these items; however, for them to have an existential value, it is not necessary that they be available to the majority of people.