I updated the Arduino code to load up the button image in the setup() function and deleted the code from the loop. I uploaded the sketch to one of the arduinos and connected it to a display. And it didn’t work.
Since it has been so long since I worked on this code, and since I’m changing its structure, I decided to go back to the original sample code from Adafruit, and get it working on my arduino uno (since that has debug support). So I did all that and it still didn’t work. The error message I got said that it failed to initialize the SD card. I pulled one of the screens out of the control panel and plugged the SD card directly into it. Now it worked. I went back to the SD extender, and it didn’t work. Then I pushed on the SD extender to make sure it was all the way in, and it worked.
It turns out that the part of the SD extender that goes into the SD socket is not quite as thick as an actual SD card. Here is a close-up photo of the adapter on top of an SD card, and you can see the difference in thickness.

I don’t think this would have mattered for most types of SD sockets, but the mini LCDs have the push-to-release socket:
And with the extender being slightly thinner than an actual card, it is not held securely in the socket:
So now I’m pretty sure that I have to disassemble the control panel (remove all the buttons, remove the plexiglass and paper) and pull each display out and then glue a thin piece of plastic onto each extender.