Concept-first: 'saturation' tells you the elimination machinery is maxed out and running flat-out at a fixed rate. A flat fixed rate cannot scale with how much drug is around.
So in zero-order elimination a constant quantity (say, mg per hour) is cleared whether the plasma level is high or low, which means the elimination rate is independent of plasma concentration, the correct option.
The other terms are not constant here. Half-life and clearance both shift with concentration in zero-order behaviour (half-life grows as levels climb), so neither is the answer. Volume of distribution describes drug spread, not the saturable elimination feature. Ethanol, high-dose phenytoin, and aspirin are the standard examples.