The segments that are not working are not the same in all four digits. For example, one digit has the center horizontal segment out, and the adjacent digit has the lower left vertical and the lower horizontal segment out. Good news I suppose since if it is the display, at least I may be able to replace it. Note that only 12 pins of the 16 pins pass through the board (two rows of 6 pins pass through). The outer 4 pins appear to be an aid for assembly since they do not pass through the board and are not connected to anything. I tried to show this with arrows in one of the photos, but there are 16 pins, with only 12 used.
I agree the original solder connections looked poor, so after I took the pictures I re-soldered them, and nothing changed.
Thanks for the tip, I will look at the digital clock kits.