Check the switch for continuity (closes circuit) with a multimeter. Check the wires for the switch to the non-functional circuit in case they got pinched in a mechanism.
If it is clear which two wires control each function at the circuit board then you can short those two pins together at the circuit board. Presumably it has live AC power? If so be mindful of that while testing.
You could identify the mechanism used to recline the chair and the motor(s) involved and measure for power at them when pressing the appropriate buttons. Also look for bent parts, any jambs in smooth movement of the mechanisms, broken springs, cracked wood, sheared gears, etc.
It may or may not help if you provide a more clear list of the movement modes, for example it is unclear to me whether backrest is a subset of recline, or the entirety of the recline and the footrest is independent, or ?? Remember only you know what this chair is supposed to do and can see all the buttons.
It may or may not help if you can provide pictures of the switch panel from the front and back and of the circuit board with indications where each switch goes.