Sprite animation on quad with UV-coordinates

I need to animate a character sprite in a 3D environment. So far, I have created a Quad, on which I have put a material containing the (possibly malfunctioning) sprite sheet/atlas (.png). I have also attached the script below to the Quad. I have two major challenges:

1. The spritesheet refuses to show up as transparent in-game. Inspecting the .png-file shows the collection of sprites with the proper transparent areas. The background is either grey or a smear of colors.

2. I have modified the script below in many ways to change the quad's current sprite by using UV-coordinates, but I can't figure out how to make this work properly. As of now, nothing changes in-game no matter what I set the UV-coordinates to.

Any ideas on where I might've gone wrong? I must've read 100 articles on UV-mapping by now.

 using System.Collections;
 using System.Collections.Generic;
 using UnityEngine;
 
 public class UVQuadMapping : MonoBehaviour
 {
     public Rect face = new Rect(0f, 0f, 1f, 1f);
     private float _elapsedTime = 0;
     private Mesh mesh;
     private new Vector2[] UVs;
     private int faceIndex = 0;
 
     void Start()
     {
          mesh = transform.GetComponent<MeshFilter>().mesh as Mesh;
          Vector2[] UVs = new Vector2[mesh.uv.Length];
          UVs = mesh.uv;
          setUVs(faceIndex);
          mesh.uv = UVs;
     }
 
     void Update()
     {
         _elapsedTime += Time.deltaTime;
         if(_elapsedTime > 1f)
             _elapsedTime = 0;
         else
             return;
 
         setUVs(faceIndex++);
          mesh.uv = UVs;
     }
 
     void setUVs(int pOffset = 0)
     {
         UVs[0] = new Vector2(0, 0);
         UVs[1] = new Vector2(1, 0);
         UVs[2] = new Vector2(0, 1);
         UVs[3] = new Vector2(1, 1);
     }
 }

Thanks! Will gladly add more details if needed.