Make vehicle automatically steer towards path

I wan’t my vehicle to automatically steer so that the player “only” has to accelerate. It turns out that this seems a bit harder than I initially thought.

Right now I have a Path defined by List<Vector3> , I get the normal of the lookAhead to the path (that’s what ``NearestPointOnLine() does). And then I rotate the vehicle accordingly and give it a relative Force forward.

I’m probably on the wrong track, so any hints or solutions in the right direction are highly appreciated.

This is what it currently looks like:

This is what I have:

        void Update() {
            FollowPath();

            if(actions.Accelerate.IsPressed){
                body.AddRelativeForce(Vector3.forward * acceleration * Time.deltaTime);
            } else {
                Vector3 vel = body.velocity;
                body.velocity = Vector3.zero;
                body.AddRelativeForce(Vector3.forward * vel.magnitude * Time.deltaTime);
            }

            body.velocity = Vector3.ClampMagnitude(body.velocity, maxSpeed);
        }

        void FollowPath() {
            // predict future location
            prediction = body.velocity;
            prediction.Normalize();
            prediction *= lookAhead;
            prediction += transform.position;

            normal = NearestPointOnLine(aimAt.waypoints[0], aimAt.waypoints[1]-aimAt.waypoints[0], prediction);

            float distance = Vector3.Distance(prediction, normal);
            if(distance > aimAt.radius) {
                SteerAtPath();
            } else {
                Debug.Log("I'm on track");
            }
        }

        void SteerAtPath() {
            NearestNormal();

            Transform desired = transform;
            desired.LookAt(normal);
            body.MoveRotation(Quaternion.RotateTowards(transform.rotation, desired.rotation, 10f * Time.deltaTime));
        }

I was just told that there’s a Standard Unity Asset with a demo of a Car following a path. I’m simply going to use that since it solves many other problems I was not yet aware of.