What to make of this strange snap dome switch configuration?
The buttons are likely configured as a scanned array. The microcontroller continuously scans the rows (or columns) and reads back the other "dimension". 

Did you look at the nodes with an oscilloscope, or just voltage readings with a meter?  It is quite common to find really tricky circuits used in these gadgets to minimize the pin count on their microcontroller to shave a few cents off the cost. 

Another possibility is using a resistor array where different buttons produce different voltages which the microcontroller can read.  None of these would be appparent from high impedance probing. But measuring the waveform or the voltage while the switch is closed may give a bigger clue to how it works.

The typical way of interfacing to an array like this is to put a FET across each switch and drive the gate from your microcontroller. That simulates pushing the button and closing the switch.