Part of the "problem" is that the system message stretching across the screen is being used for routine messages. The "system message" was designed to be used only under extraordinary circumstances… and it's supposed to really stand out. That drowns out the tab block just underneath it.
Re-designing the top-right tabs to out-scream the screaming system message isn't really a good answer.
Moderators are free to set the system message and many are using it to publicize routine aspects of the site. But when you blast out a message on every single screen across the entire site, that's just too much. We simply need a better way to provide system-wide announcements/messages so they don't have to use the Stack Exchange-equivalent of the Emergency Broadcast System.