But that said, my understanding, very likely wrong, was that those were mostly tax records and other lists - which don't fire my imagination in quite the same way as works of philosophy and literature snatched (almost literally) from the flames of history.
Now, why should I be more interested in the mesopotamian tablets? (Not sarcasm, I'm interested)
Remarkably, these figures and their writings, dating from ~2300 BC, were as distant from Julius Caesar as he is to us, and yet they played a major role in shaping our world, for instance by setting the early foundations for Judeo-Christian thinking (examples: the flood story, Enheduanna's laments, etc). So we have every reason to be interested in them.
It would, of course, be great to do both. But my point is that it is going to be much harder to attract funders, participants, press coverage, and so on for reading Mesopotamian tablets than for reading Greek or Roman papyri excavated from Piso’s villa in Herculaneum.