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**How would I go about making these thin lines in Blender appear in Unity? **
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You can convert them to basic meshes (before exporting), this is the easiest approach.
Or you can try changing MeshTopology from script. But mesh’s indices must follow an actual line structure { 0,1, 2,3, 4,5, ... } or { 0,1, 1,2, 2,3, ... }, because otherwise it will render just a web of lines.

using UnityEngine;
public class SetMeshTopology : MonoBehaviour
{
[SerializeField] MeshTopology _meshTopology = MeshTopology.Triangles;
[SerializeField] MeshFilter _meshFilter = null;
[SerializeField][Min(0)] int _submesh = 0;
[SerializeField] bool _sharedMesh = true;
void Awake ()
{
var mesh = _sharedMesh ? _meshFilter.sharedMesh : _meshFilter.mesh;
var indices = mesh.GetIndices( submesh:_submesh );
mesh.SetIndices( indices:indices , topology:_meshTopology , submesh:_submesh );
}
}
