I do it, it's probably my main playstyle. I'm a completionist to a fault! You can go through one tier of all of the careers, save the sim in CAS, then start the CAS sim over again to complete the "other side" of the career branches.
A lot of the traits give RNG mood swings that often interfere with what I'm trying to do, so I try to avoid those as much as possible. I personally like to use the Athletic one to keep them from getting fat so I don't have to worry about stuff like calories while I've got my sim slaving away at skills.
The other traits I usually default to are Carefree, Bookworm, and Perfectionist, because you don't get any annoying bad moodlets from those. If they changed it so that you did, I probably would quit for sure and never waste my time in this game again. But it works for now, I've done a number of sims that way over the years. You really can do all the careers and aspirations on one Sim.