Turns out that I have over-engineered this shield a little bit — I wanted to make sure the pins are pulled up when the buttons are not pressed, so I added a pull-up resistor to each button. But looking at the #LAMEBOY - another ESP12 handheld project, I noticed that @davedarko didn't use any pull-ups, and it works reliably for him. We dived into the datasheet a bit and did some experiments, and it turns out you really don't need them. So now I'm leaving the pads for the pull-up resistors un-populated. Perhaps I will design another version of the PCB, with the chip on the bottom side and without any pull-ups whatsoever, but for now this simplified second version works well for me.