Why does using a transistor as a switch not work?
Transistor need a certain voltage and polarity that can be switched. In addition the controller signal must have right polarity and ground, the ground should be one of your switchboard pads. It's well possible that right polarity common voltages and ground do not exist. Without a schematic and without having your toys in my hands it's impossible to say does a way to place a transistor switch exist and what it is (see NOTE1). If there's no schematic some oscilloscope measurements may bring the light if the circuit functions in simple enough way.
Relay allows a large range of voltages and currents and it has totally separate in and out. Use it if it works otherwise well enough. Optocoupler have the same separation property, but the polarity of the voltage is critical if the optocoupler has transistor output. LDR output optocoupler very likely works as well as a relay, but with no moving parts.
NOTE1: Matrix type multi-button interface has neither of the plates = GND or operating voltage. Then there's a button reading principle which uses AC signals. It's based on capcitance change. The ultra thin PCB stripes hint (but not prove) that the button reading principle isn't the simplest possible.