My source here is the _Nihon Kokugo Dai Jiten_ entry here at Kotobank, where the verb is the fifth section down, starting with the line ().
According to this, the verb is first cited to a text from roughly 1001 with an unvoiced middle consonant as . I think this bolsters the idea that this is a verb derived from the noun (spelled variously in kanji as etc.). The voiced appears later, apparently in the 1200s or 1300s. My suspicion is that the bilabial nasal //m-// at the start of the word may have gradually caused voicing of the following //-t-// to result in voiced //-d-//. There may also have been a need to differentiate from homophonous verb with the same []{LHL} pitch pattern, but a different meaning ("to bend something, to warp something").