You are right in identifying that this essay includes rather a gruesome and, arguably, unnecessary description of how the poor elephant that Orwell did not actually want to kill met his end at Orwell's hand. Note the following rather graphic description:

I fired a third time. That was the shot that did for him. You could see the agony of it jolt his whole body and knock the last remnant of strength from his legs. But in falling he seemed for a moment to rise, for as his hind legs collapsed beneath him he seemed to tower upward like a huge rock toppling, his trunk reaching skyward like a tree. He trumpeted, for the first and only time. And then down he came, his belly toward me, with a crash that seemed to shake the ground even where I lay.

We definitely feel sympathy for the element, but also, let us remember the real reason for such a protracted description: it helps us feel greater sympathy to for Orwell in the impossible position in which he finds himself. For Orwell himself says he was not a good shot nor a good hunter, and he did not want to kill this animal that he was expected to slaughter for a public display in front of the Burmese that despise him so.