There is another aspect of working on big projects that seems equally important. What you are talking about I'd call "design", the skill of organizing the code (and more generally, the development process) so that it remains intelligible and easy to teach new tricks as the project grows. It's the kind of thing reading SICP and writing big things from scratch would teach.

The other skill is "integration", ability to open up an unfamiliar project that's too big to understand well in a reasonable time, and figure out enough about it to change what you need, in a way that fits well into the existing system. This requires careful observation, acting against your habits, to conform to local customs, and calibration of the sense of how well you understand something, so that you can judge when you've learned just enough to do your thing right, but no less and not much more. Other than on a job, this could be learned by working a bit (not too much on each one, lest you become comfortable) on medium/large open source projects (implementing new features, not just fixing trivial bugs), possibly discarding the results of the first few exercises.