How do you pick up an object with a tag? [must have physics and no inventory]

Ok, I made an Amnesia clone a while back using coroutines like this for rigidbody interaction so this should behave how you wanted, it pulls objects from their center rather than a single point like with the spring joint method. It just needs an empty parented to the camera.

 public Transform player_camera;
 public Transform phys_dragger; //empty parented to camera
 public float grab_distance = 5.0f;
 private float temp_drag = 10.0f;

 IEnumerator DragObject(Rigidbody r){
     float oldDrag = r.drag;
     r.drag = temp_drag;
     r.useGravity = false;
     while(Input.GetButton("Interact")){
         //Keep aligned with camera's view
         r.transform.rotation = Quaternion.Slerp(r.transform.rotation, phys_dragger.rotation, Time.deltaTime * 15.0f);
         //Vector that points from rigidbody to grab point
         Vector3 vel = (phys_dragger.position - r.position).normalized;
         //Clamp distance to 1, otherwise it could be pulled through walls
         float d = Mathf.Min(Vector3.Distance(r.transform.position, phys_dragger.position), 1.0f);
         r.AddForce(vel * d, ForceMode.VelocityChange);
         
         if(Input.GetButton("Throw")){
             r.rigidbody.AddForce(player_camera.transform.forward * 5, ForceMode.Impulse);
             r.drag = oldDrag;
             r.useGravity = true;
             yield break;
         }
         yield return new WaitForFixedUpdate();
     }
     r.drag = oldDrag;
     r.useGravity = true;
 }

Then it would be used with a raycast so you can pass it the rigidbody that was hit without needing a permanent variable to store it in.

 void Update () {
     if(Input.GetButtonDown("Interact")){
         RaycastHit hit;
         if(Physics.Raycast(player_camera.position, player_camera.forward, out hit, grab_distance)){
             if(hit.rigidbody){
                 phys_dragger.position = hit.rigidbody.position;
                 phys_dragger.rotation = hit.rigidbody.rotation;
                 StartCoroutine(DragObject(hit.rigidbody));
             }
         }
     }
 }

You'll want to mess around with the temporary drag and forces, if you lower the drag you'll notice it will orbit around the point but if it's high the object will move too slow.