Method not working inside FixedUpdate

I'm not sure how much of a performance hit finding objects of a specific type in a scene is but if you know for a fact that there is only ever going to be one GameManager within a scene you could setup the GameManager class like this:

 public class GameManager : MonoBehaviour
 {
     public static GameManager instance;
     
     Awake()
     {
          if(!instance)
              instance = this;
     }
 }

That way you can call GameManager without having to do any searching at all, simply doing GameManager.instance.EndGame();

as for the issue of not being able to call the method from the FixedUpdate method, I'm not entirely what the issue is, I've used the same code on many occasions and was able to do things like get components and call methods like it was a regular GetComponent call.

I think you should do

GameObject gameManager = GameObject.FindObjectOfType<GameManager>().EndGame();

in Update() or FixedUpdate() of course.