It would seem the language is understood quite well, but the cultural context of its use not so much.
Akkadian is a Semitic language, quite distantly related to Arabic and Hebrew, etc. So we know what its siblings turned into, and we have a vague guess for what its unwritten ancestor was like. There are also some surviving bilingual and trilingual Akkadian texts especially in the later period. That includes with Ancient Egyptian, Ancient Greek, Old Persian, Aramaic, etc., which are all understood to varying degrees.
It's interesting to compare to Sumerian, the language of the people who the Akkadians learned writing from. It's not related to anything else known. We only know it through Akkadian bilingual texts, and texts teaching Akkadians how to read and write Sumerian. (Sumerian went extinct by around 2000 BC.) Now Sumerian is much more poorly understood.
Tablets is are sometimes found when cities fell or buildings were abandoned, basically frozen in context. In this case, the letters were excavated from one of the large homes in Sippur near the temple, which had three rooms full of clay archival tablets seemingly related to the life of an important, sometimes-wealthy priest-administrator, descended from a line of them. Chief dirge singer was indeed some priestly office related to public performances of ritual song.
My very small awareness of the history made it sound fairly speculative.