Why "job" in "nut job"

Maybe looking at its origin will help. SlangDictionary.com defines "nut-job" as:

a more recent variation of nut-case

"Nut-case" being:

a crazy person. A slang version of ‘mental case’ which spread from American speech into British usage at the end of the 1950s.

Thus, if "nut-job" was a mere variant of "nut-case", then, "job" would make sense. A "case" has two meanings:

a person or thing whose plight or situation calls for attention:
A piece of work, specifically defined within a profession.

The second meaning of "case" is synonymous with "job", and thus, in when "nut-case" was given a variation, its synonym "job" was taken and used. Thus, there would be a logical reason why "job" is used here.

This use is very similar to whack job and also similar to other slang uses of job such as "do a job on" or "a bank job" (meaning to rob a bank). Here, job just means something of a particular type.

"I got a new oven, one of those self-cleaning jobs."

"Use a real-dictionary, not one of those new-fangled online jobs."

You'll also find nut job used a lot in this same construction. "He's one of those right-wing conspiracy nut jobs."

Job here is used informally to refer to something of a certain kind. See for example the definition under object in the OALD or the definition under example in the CALD.

This one may be more closely related to "jobber," which means a "person who does a particular job," but also gained an extended sense of "guy," or "fellow," e.g., "See that jobber over there." Over time, jobber may have been truncated to "job."

I knew the etymology of "nut" from OED in that it can refer to the head. Etymology of "job" refers to "piece of work" from the 1550s (http://www.etymonline.com/index.php?term=job). I would guess the way that we slangily use "piece of work" to mean someone who doesn't make sense, "nut job" would logically follow as it is shorter than "nut piece of work"...