I do not find that the sentence is grammatically incorrect in first place. Also the official explanation could not convince me that there is grammatical mistake in the sentence.

Based on accounts of various ancient writers, scholars have painted a sketchy picture of the activities of an all-female cult that, perhaps as early as the sixth century B.C., worshipped a goddess known in Latin as Bona Dea, "the good goddess."

(B) Basing it on various ancient writers' accounts,
(C) With accounts of various ancient writers used for a basis,
(D) By the accounts of various ancient writers they used,
(E) Using accounts of various ancient writers,

Please explain. the OA is (E)