Combining two meshes

I'm working on making some of my own meshes in runtime. Just simple things like boxes and cylinders, but I'm having a little trouble. I want to simply combine a cylinder and cone to make an arrow, but when I try I get an empty game object. Here's what I'm trying to do.

   public static Mesh CombineTwoMesh(Mesh mesh1, Mesh mesh2)
   {
     CombineInstance[] combine = new CombineInstance[2];
     combine[0].mesh = mesh1;
     combine[1].mesh = mesh2;
 
     Mesh newMesh = new Mesh();
     newMesh.CombineMeshes(combine);
 
     return newMesh;
   }

Where I call this basically with my arrowMesh and coneMesh. I hope to expand upon it with the transforms in CombineInstance, but I can't seem to get it to work this way. Any help or tips would be appreciated.

BTW, I know I could do this with stacked gameObjects, but I'm trying to avoid as much hiearchy as I can for the intended simple uses I'm perceiving here.