As jargon for an unsolicited sales pitch, cold call, was used way before the 1970s. The earliest use I can find is from Volume 100 of The American Magazine in 1925:
. . . I do need insurance.' He signed up forthwith for five thousand dollars. "I suppose you might call that opportunity a 'hunch.' I had no introduction to the man, no personal link of any kind. It was a cold call, and it won.
This snippet is all I can get through Google. If anyone can access this magazine, it would be great to get more context for the quote. The term seems to have been popularized thereafter by salesmen trade publications in the later 1920s.
The phrase itself is most likely older than this. I have two other citations where the meaning of its use is not exactly clear to me. The first is from a collection of field notes
taken by British entomolgist Augustus Radcliffe Grote and published in 1877:
I also had visitors still higher up in the scale of nature. Some Indians, from the Reservation near by, paid me a cold call. These did not come to 'sugar,' reconnoitring perhaps for whisky.
This seems to suggest that the phrase may have formed as a variation of the phrase pay a call (in use since the early 1800s and itself a variation of pay a visit--in use since at least the mid-1600s). But it could also just be describing the demeanor of the Indians.
The second is substantially earlier and is from Vol. XII of The Oriental Herald, and Journal of General Literature, 1827:
The same feeling forbade to wait the result of a cold call for Colonel J. W. Adams's services ; and the staff of the army was completed before that veteran could recal his leave of absence on account of sickness.
Again, it is unclear what exactly is meant by cold in this citation. It could mean the call was calculated and unfeeling, but it could also imply that it was unexpected.
As Robusto comments, it's probably just an obvious turn of phrase that could easily have been "re-coined" repeatedly.
As Google NGrams shows, the term had no real currency until the 1980's, but that certainly doesn't mean such bands didn't exist (see this site).
Personally I think that it's quite possible many parents of wannabe rockers (esp. perhaps West Coast Americans) may have used it to describe what their kids get up to on the weekend, even back in the 50's. But it didn't really gain traction until the post-punk era, when the type of music associated with it started to become commercially significant.
Best Answer
The earliest attestation I can find ascribes the origin to Forel's wortsalat, as translated by Kraepelin to 'wordsalad'. This appears in The Medical Standard of 1895, recounting events from the May, 1894 meeting of the Association of German Alienists and Neurologists, where "Kraepelin of Heidelberg described a 'peculiar group of insane patients,' who, among 'other distressing symptoms,' exhibited as 'the most striking phenomenon' a tendency to the coinage of new words":
Although the earliest attestation in OED Online is from 1904, they give the origin as "after either German Wortsalat (1894 or earlier) or French salade de mots (1895 or earlier)", which jives with the date of the wortsalat attestation in The Medical Standard.