There's another episode near the end of Enterprise, where Dr. Phlox is kidnapped to help a Klingon doctor make Klingon augments, using Dr. Soong's augment's DNA. It's pretty clear in that episode that, while there are Klingons that pursue other professions, they aren't thought of very highly by the warrior caste.

Also, they have a caste system, so there are definitely Klingons that have to do other jobs, and this likely would have been even more important in the distant past. General Martok in DS9 had difficulty rising through the ranks originally, because he wasn't from the warrior caste