As mentioned in Emil Jeřábek's comment (which quickly gained several upvotes), at least some of the "network-wide stuff" (Meta Stack Exchange posts, blog posts) can be useful also for MathOverflow users.
Let me suggest an alternative proposal to the complete removal of those links from the community bulletin.
Suggestion. ReorderingReorder the links in the community bulletin in such way that the posts which have importance for MathOverflow (posts from MathOverflow Meta, election announcements, etc.) would be at the top and posts originating from Meta Stack Exchange (and blog posts) would be below them.
If the main concern is that the posts from Meta MathOverflow get less attention because they are at the bottom, this might help a bit. This was a suggestion which was proposed by Meta Andrew T. when I recently asked on Meta Stack Exchange about another suggestion related to the community bulletin: Allow local mods to remove items from community bulletin. (Perhaps the answers posted there are worth reading in connection with this discussion.)
I suppose that it might make sense to post a proposal like this on Meta Stack Exchange. However, I thought it might be reasonable to see what MO users think about this - since a discussion about community bulletin started on this meta anyway. I do not plan to post two feature requests concerning the community bulletin on Meta Stack Exchange so soon after each other - but if somebody else decides to make a feature request along these lines, I certainly have no objections to that.