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:

Code (CSharp):

1.         void Update() {
2.             FollowPath();
3.  
4.             if(actions.Accelerate.IsPressed){
5.                 body.AddRelativeForce(Vector3.forward * acceleration * Time.deltaTime);
6.             } else {
7.                 Vector3 vel = body.velocity;
8.                 body.velocity = Vector3.zero;
9.                 body.AddRelativeForce(Vector3.forward * vel.magnitude * Time.deltaTime);
10.             }
11.  
12.             body.velocity = Vector3.ClampMagnitude(body.velocity, maxSpeed);
13.         }
14.  
15.         void FollowPath() {
16.             // predict future location
17.             prediction = body.velocity;
18.             prediction.Normalize();
19.             prediction *= lookAhead;
20.             prediction += transform.position;
21.  
22.             normal = NearestPointOnLine(aimAt.waypoints[0], aimAt.waypoints[1]-aimAt.waypoints[0], prediction);
23.  
24.             float distance = Vector3.Distance(prediction, normal);
25.             if(distance > aimAt.radius) {
26.                 SteerAtPath();
27.             } else {
28.                 Debug.Log("I'm on track");
29.             }
30.         }
31.  
32.         void SteerAtPath() {
33.             NearestNormal();
34.  
35.             Transform desired = transform;
36.             desired.LookAt(normal);
37.             body.MoveRotation(Quaternion.RotateTowards(transform.rotation, desired.rotation, 10f * Time.deltaTime));
38.         }
39.  

 

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.