Well, it is important to remember that in this excellent essay the elephant of the title is a powerful symbol that is used to support the author's thesis statement. If we want to pick out that thesis statement, we would need to look at the following quote:
I perceived in this moment that when the white man turns tyrant it is his own freedom that he destroys.
Orwell discovered through the incident when he was forced to shoot the elephant the truth of this statement. He, apparently, was the man in charge, the man with the power, and yet, he realises that he had no choice but to shoot the elephant even though he felt that it didn't need to be killed and he didn't want to do it. Ironically, becoming a tyrant has actually resulted in limiting and restricting the freedom of white men, rather than increasing it.
Thus, you are right in a sense when you say that the elephant has nothing to do with the thesis statement, however it was the incident that triggered the powerful epiphany that Orwell experienced about the colonial exploits of the "white man." In a sense, there could have been a number of incidents that would have sparked that same sudden understanding, but Orwell chose to base it on his own shooting of an elephant.