This certainly is an interesting idea, but, in my opinion, I just don't think it would work given most situations. As it is, most people will just call for engineering in general to come fix any of their issues. To have this work well, and not just be a novelty job, I'd imagine engineers would have to be revamped and have their basic construction supplies taken away. Essentially a new sub-department would be created with the goal being construction around the ship. It's an interesting idea for a job, but I think it would take some tweaking for anyone to care about it.
And to having to work on things one may be unskilled with, all I have to say is practice makes perfect. Can't be good at it until you have to do it a few times