Why is a normal bike's front tire contact patch often behind the steering axis?

Answer: It lets road forces act behind the steering axis so the front wheel tends to self-center

It puts more tire behind the steering axis so hard braking presses the contact patch into the road — Wrong — trail is steering geometry, not a braking-grip feature, and the contact patch position mainly changes how road forces steer the wheel. It does affect how load reaches the tire, so braking feel can change as a side effect. The lever arm around the steering axis is what makes the wheel tend to align itself.

It lets road forces act behind the steering axis so the front wheel tends to self-center ✓ — Correct — the backward offset is trail, and it gives road forces a lever arm around the steering axis that tends to align the front wheel with the bike's direction of travel. That self-centering helps the bike track straight and recover after small disturbances. It is a stability feature of the steering geometry, not a trick for sharper low-speed turning.

It shortens the steering arm to the contact patch so low-speed turns need less handlebar movement — Wrong — the offset can change handling feel, but its main job is not to make low-speed steering quicker. Trail makes the front wheel tend to follow the direction of travel and settle after a disturbance, which is a stability effect. A shorter trail would feel quicker, but the usual backward contact patch is there for self-centering.