As you can see from the updated gallery photos, I have a working binary clock. To review, the minutes and seconds are 1, 2, 4, 8, 16, 32 from left to right, top to bottom, and the hours are in a line at the bottom 1, 2, 4, 8, 16 from left to right, displaying 24 hour time; 13:42:17 in the picture. The firmware is a straightforward initialization of the STM32's built-in battery backed RTC followed by some simple LED multiplexing. Unlike my older clock, this clock will keep excellent time with a 32kHz clock crystal, even if it is left unplugged for a year or two. The case is more like a stand, a quick and easy 5mm plywood front panel glued to an angled side of a piece of 3/4" plywood, painted light blue.
This was a fun little project. Just sitting down and building something, no new boards, easy firmware. Eventually should add finishing touches like shortening the wires, screwing everything down, maybe enclose the back, but for now I'll enjoy my new combination clock-nightlight.