Aha! Turns out my camera isn't broken, and doesn't have two dead batteries. The clock display is some weird Easter egg mode that I must have tripped by accident, and IT DOESN'T RETURN TO NORMAL EVEN WHEN YOU TURN THE CAMERA OFF AND BACK ON.
https://support.usa.canon.com/kb/s/article/ART154539
Well, it's supposed to, so maybe there is something broken in my camera.
I have tripped clock mode before, but never deliberately, and it always used to clear up after a few seconds (like the manual says it should). Today's the first time it actually got stuck. Nothing I tried would bring it out and let me, you know, take pictures with it.
Fortunately, the Power+Func/Set combo does still work to toggle back out of the clock mode, so maybe next week I can actually take some photos of birds that look like birds and not just blobs in the distance.