The scholarly and the popular receptions of this text have run on different tracks, and the gap between them is instructive.
The Sumerian hymns attributed to Enheduanna were edited and published in a scholarly edition in 1968 by William Hallo and J. J. A. van Dijk, which is when the material became usable. For two decades the attribution question was discussed within Assyriology in the ordinary way.
From the 1980s the figure was taken up outside the field, in feminist history, in poetry, and in popular writing about goddess religion, where she is regularly introduced as the world's first named author with the qualifications dropped. Anthologies of women's writing open with her. A crater on Mercury is named for her. Exhibitions have been mounted around her. The reception is not baseless, the case for the attribution is respectable, and the disc from Ur is real, but it has generally passed over the fact that the manuscripts are half a millennium later than the person.
Both things can be held at once, and the honest formulation is available: a body of Sumerian hymnic poetry ascribed in antiquity to a historical high priestess, preserved in copies from centuries afterward, whose authorship is probable to some specialists and doubtful to others.
The reason to be careful is not that the claim is grand. It is that the same evidentiary standard, applied to a male figure, would produce the same caveat, and applying it unevenly does the subject no favours.