In a large shop you may work in one specialization. For example, you may just replace glass or do painting. So it may be possible to be hired without knowing the full range of skills. Still, a broad training program increases your chances of finding work.

You may find it helps your career to be certified by the National Institute for Automotive Service Excellence (ASE). This requires you to take at least one exam of the four they offer and have two years of experience. (Postsecondary education in this field can substitute for one of those years.) Note that you have to retake the exam after five years to stay certified. ASE recognizes that you have to keep your skills up-to-date.