I've been learning how to use a DC motor driver board, and came across flyback diode in the description. I've read about its importance to limit or prevent the formation of the electric arc across the switch contacts. But is that the only protection it offers? So, without a flyback diode across an inductive load, like a motor, wouldn't the electronics of the rest of the circuit also be at risk of voltage spike damage?
Define "electronics of the rest of the circuit". Normally, if a switch is controlling the motor there is nothing else connected when the switch goes open. But if that circuitry is switching to motor (H-bridge/motor driver) then yes, it protects them as well.
Yes, arcing on a switch generates high frequency high voltage energy that both damages electronics and gives out lots of radio interference. High voltages can cause breakdown of the insulation on the windings in the motor itself, so it becomes unusable. And of course with a large motor the voltages without a diode can be lethal even if the supply is 48V or less.