> If it broke, don't pull the project with the breaking changes.
I agree with the overall sentiment of your comment, but the quoted part is where I've seen trouble brew. The tendency is to be conservative about pulling updates to dependencies, which can easily get you into a very awkward state when a critical update eventually sits on top of a bunch of updates you didn't take because they broke you. It is usually better to be forced to handle the breakage immediately, one way or another.
I don't see how this is a problem if you are pushing frequently and have a CI system. You know within minutes if the build is broken. If it broke, don't pull the project with the breaking changes.
My point is, I don't think one approach is inherently better than the other. Both require effort on the part of the teams to manage changes (or a CM team), and both require defined processes.