Lightning Using A Line Renderer

So far I have created a line renderer that follows a bezier curve: I am now trying to make the line look like lightning. I have tried Perlin noise and it didn't really work as I expected: https://imgur.com/a/PUhfWn5

Each point sort of jerks around wildly and it looks odd. I want the effect to scroll like a texture if that makes any sense.

Here is my code:

 [SerializeField] Transform point1;
 [SerializeField] Transform point2;
 [SerializeField] Transform point3;

 [Min(1)] [SerializeField] int vertexResolution = 12;
 [SerializeField] Vector2 noiseScale;

 LineRenderer lineRenderer;

 void Start() => lineRenderer = GetComponent<LineRenderer>();

 void Update()
 {
     List<Vector3> pointPositions = new List<Vector3>();
     for (float i = 0; i <= 1; i += 1f / vertexResolution)
     {   
         Vector2 tangentLineA = Vector2.Lerp(point1.position, point2.position, i);
         Vector2 tangentLineB = Vector2.Lerp(point2.position, point3.position, i);
         // Each point along the curve
         Vector2 bezierPoint = Vector2.Lerp(tangentLineA, tangentLineB, i);

         // Checks if the current vertex is the first or the last, which makes both ends of the lightning stay still.
         if (i >= 1f / vertexResolution && i <= 1 - 1f / vertexResolution)
         {
             // Creates noise and maps it from 0 - 1 to -1 and 1.
             float noise = Mathf.PerlinNoise(Time.time, 0) * 2 - 1;

             // Changes the point's position randomly based on the noise.
             Vector2 point = new Vector2(bezierPoint.x + noise * noiseScale.x, bezierPoint.y + noise * noiseScale.y);
             pointPositions.Add(point);
         }
         else
         {
             pointPositions.Add(bezierPoint);
         }
     }t

     lineRenderer.positionCount = pointPositions.Count;
     lineRenderer.SetPositions(pointPositions.ToArray());
 }