We're back to Status="Yay" again. A call to Zune tech support resolved the issue. We went through the normal checklist of things to try, most of which I'd already done prior to calling in. (Such as testing multiple headphone sets, checking different audio sources like radio and recorded music, etc.)
But the one thing that I hadn't tried yet, and it's not at all obvious on the device how to do this, was a reset. It involves hitting two buttons simultaneously and holding them for five seconds. THAT worked. So, like I said, we're in "Yay" mode again.
Good. I could use some "Yay" mode.