I have a lot of free play too...and my wife bought herself another car (SX4 AWD) because she could not handle it.
Power steering always have a dead play at rest (engine not running) because the initial movement of the steeering shaft is the power steering control valve input. However, as soon as the engine runs and the pump supplies pressure, a tiny amount of valve action opens the oil flow and this is what drives the pitman arm...not your direct steering input, and you think there is no free play because the PS does the job instantly.
If valves become sticky or obstructed, oil flow is retricted and you have to turn the steering more and eventually until it reaches the end of the valve free play and then YOU turn the wheels. Use of unsuitable steering fluid can dissolve valves and seals and degrade or ruin the power steering action. So....it turns out indeed that the gearbox..containing the PS control valves...is shot...or your power steering pump is shot.