Make Vertices selectable and shown as UI

Creating thousands of UI elements to represent all the mesh vertices just to make them selectable is a bad idea. Full stop.

Drop it and move forward with something much more simple, reliable and performant. The very first thing you should try is to just calculate vertex selection 100% manually.

How to do that? Just iterate over all vertices and look for Ray & sphere or screen point & Rect intersections on mouse click. If performance starts to be the issue for some reason - no worries, there is plenty of space for optimizations here (AABB sets, hierarchies etc).

 using UnityEngine;
 
 [RequireComponent( typeof(MeshFilter) , typeof(MeshRenderer) )]
 public class LetsSelectMeshVertices : MonoBehaviour
 {
     
     [SerializeField] Camera _camera;
     [SerializeField] float _hotSpotSize = 30f;
     [SerializeField] int _selectedVertexIndex = -1;
     Vector3[] _vertices;
 
     void Start ()
     {
         _vertices = GetComponent<MeshFilter>().sharedMesh.vertices;
     }
     void OnBecameVisible () => enabled = true;
     void OnBecameInvisible() => enabled = false;
     void Update ()
     {
         if( Input.GetMouseButtonDown(0) )
         {
             Vector2 mousePosition = Input.mousePosition;
             Matrix4x4 ltw = transform.localToWorldMatrix;
             for( int vertexIndex=_vertices.Length-1 ; vertexIndex!=-1 ; vertexIndex-- )
             {
                 Vector3 vertexWorldPos = ltw.MultiplyPoint3x4( _vertices[vertexIndex] );
                 Vector3 vertexScreenPos = _camera.WorldToScreenPoint( vertexWorldPos );
                 Rect vertexScreenRect = new Rect( vertexScreenPos.x-_hotSpotSize*0.5f , vertexScreenPos.y-_hotSpotSize*0.5f , _hotSpotSize , _hotSpotSize );
                 if( vertexScreenRect.Contains(mousePosition) )
                 {
                     _selectedVertexIndex = vertexIndex;
                     return;
                 }
             }
             _selectedVertexIndex = -1;
         }
     }
     #if UNITY_EDITOR
     void OnDrawGizmos ()
     {
         Matrix4x4 ltw = transform.localToWorldMatrix;
         
         if( _vertices!=null )
         for( int vertexIndex=_vertices.Length-1 ; vertexIndex!=-1 ; vertexIndex-- )
         {
             Vector3 vertexWorldPos = ltw.MultiplyPoint3x4( _vertices[vertexIndex] );
             Gizmos.color = Color.blue;
             Gizmos.DrawSphere( vertexWorldPos , 0.05f );
         }
 
         if( _selectedVertexIndex!=-1 )
         {
             Gizmos.color = Color.red;
             Gizmos.DrawSphere( ltw.MultiplyPoint3x4(_vertices[_selectedVertexIndex]) , 0.05f );
         }
     }
     #endif
 
 }