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.
OED
The OED defines this bag as:
c. fig. A preoccupation, mode of behaviour or experience; a distinctive style or category; esp. a characteristic manner of playing jazz or similar music. Cf. bag of tricks at sense 18a. slang (orig. U.S.).
1960 J. Hendricks in D. Cerulli et al. Jazz Word (1962) 140 Lack of acceptance is a drag... Man, that's really in another bag.
1962 Jazz Jrnl. Mar. 30 ‘Bag’ is a current piece of trade jargon for hip musicians, and means something between a personal style and a body of work.
They further say:
(not) to be one's bag slang (orig. U.S.): (not) to match one's personal style, taste, or preference; (not) to form part of one's interest, preoccupation, or area of expertise. Usually in negative contexts. Cf. thing *n.*1 4d.
1966 N.Y. Times 20 Nov. d13/2 They were trying to categorize me..as a racial satirist, but that's not my bag. Let's say I deal in universal human foibles.
1961 examples
I found some 1961 examples of this sense of bag.
Billboard magazine (6 Nov 1961) contains the following in a list of new LPs to be released by Verve. Cal Tjader was a Latin jazz musician and Verve Records is an American jazz record label
STAN GETZ AND BOB BROOKME YER— V-V6-8418 (Nov.)
IN A LATIN BAG— Cal Tjader— V-V6-8419 (Nov.)
THE TRIO— Oscar Peterson, Ray Brown, Ed Thigpen— V-V6-8420 (Nov.)
BUDDY RICH BLUES-CARAVAN— V-V6- 8425 (Nov.)
Billboard of 20th November 1961 has an advert from Verve:
IN A LATIN BAG-Cal Tjader's torrid group in a program of crackling Latin-American jazz. Sensational sound!
And Billboard of the following week (27 Nov 1961) has a review:
IN A LATIN BAG Cal Tjader. Verve V 8419— Here's a warm and winning jazz set, which which combines Latin rhythms with jazz, and does it stylishly. Cal Tjader, with Armando Peroza, Paul Horn, Al McKibbon, Wilfredo Vicent, Johnny Re and Lonnie Hewitt, handle the charts with ease, and the disk marks a noteworthy debut for Tjader on the label. The tunes are mainly originals, sparked by by Tjader's "Davito" and "Paunetos Point," plus good readings of "Speak Low," and "Ben Hur," from the current flick. Lucid, meaningful jazz here.
A snippet of Down beat magazine dated 1961 includes a review of Tina Brooks' True Blue album, released in 1960:
Soul is appropriately earthy, medium tempoed, and melodically a bit doubtful as to what jazz bag it belongs in.
Best Answer
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:
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:
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:
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.