Why does the front wheel of a leaned motorcycle often seem to find a useful steering angle without the rider holding it rigidly?

Answer: The front tire's contact patch sits behind the steering axis, so a lean steers the wheel into the turn.

The rider keeps the handlebars straight and rigid. — Holding the front wheel rigidly straight would block the small steering response described in the question. A leaned motorcycle follows a curve with a subtle steering angle. The useful feeling is not a frozen handlebar; it is a front end that can respond.

The front tire's contact patch sits behind the steering axis, so a lean steers the wheel into the turn. ✓ — Correct. In common front-end geometry, the tire's contact point sits behind the steering axis, so a lean tends to steer the front wheel into that lean. This is why a bike can feel as if it is helping find the turn. The useful feeling comes from front-end geometry allowing small steering responses, not from the rider freezing the bars.

The rear tire pushes the front wheel into the correct angle. — That answer does not explain why the front wheel finds its own useful angle. This question is about front-end self-steering. The helpful sensation comes mainly from steering geometry and the small front-end movements it encourages.