Why did Saturn V have fins?
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Those things are tiny. Surely the attitude authority of the gimbaled F-1s was more than sufficient to control the vehicle. At a minimum, I can only imagine the fins being relevant around maybe max Q?
Suppose a large launch vehicle such as Saturn V has a serious autopilot failure at the most critical part of its ascent through the atmosphere the "high-stagnation-pressure" period when the speeding rocket bucks the most severe aerodynamic forces. A failure in a swivel actuator may throw one of the five booster engines into a "hard-over" deflection, while an additional electrical failure may prevent the other engines from counteracting the unwanted turning moment.
In such a case, if high inherent aerodynamic instability assisted in rapidly increasing the angle of attack, structural overload might break up the rocket before the astronauts in the Apollo Command Module, triggering their escape rocket, could put a safe distance between themselves and the ensuing fireball in the sky.
