"Specialists get farther than generalists." I think this is a statement of very dubious veracity.
I'd rather someone be deep that shallow. In my career I've found that people who have unique talents tend to rise to the top because they offer that unique something that others dont.
Now, that unique talent can also be something broad like being a general, or being the the guy whose best at code reviews.
My point is that you've got to be really good at something, not just kinda good at a lot of things.
Just because you are the absolute best at one thing does not mean you are incompetent in all other areas.
To take your example: of course you wouldn't want to promote someone who only knows the code of one project very well and of course you want to promote someone who understands how all the parts of the product work together. However, the person who knows how all the parts work together and is the absolute best at shipping code, get a lot farther than the person who knows how they all work together, but doesn't execute any given function well enough to handle on his own