The speaker of this poem seems like the kind of guy you'd want to have on your team if you were a Survivor contestant. He seems to know stuff: "It is usually best to roll them into the canyon" (3); "My fingers touching her side brought me the reason" (9). And he keeps pretty calm, even in the face adversity and death. Given Stafford's background (he was a guy who loved the outdoors and didn't mind getting his hands dirty), it seems reasonable to consider some overlap between the poet as the speaker, though it's always dangerous to confuse the two completely. Here, we might say that the speaker is at least a man with similar knowledge and experience.
One of the first things we notice about this speaker is that he tells the story of the dead deer and his task in an even, controlled way. He doesn't get dramatic or emotional or sentimental when he tells us this tale. This is important. Imagine the poem with a very emotional or overly dramatic speaker—it turns into a very different, probably very sentimental, poem. The poem needs a very even, controlled speaker to balance the drama and emotion of the situation. When things are all one way, all one note, they tend to be less interesting. It's like that friend you have that is always drama, drama, drama. Pretty soon, you start to tune them out. The drama just doesn't seem as urgent anymore. Now consider the poem: dead animal. Dead, cute animal. Dead, cute animal and baby dead, cute animal. Stafford knew a lot of heartstrings were getting tugged all at once in this one. A weepy speaker probably would have been one too many. He needed that steady speaker as a counterpoint to the emotional events in the poem.
In fact, the speaker kind of moves into the background, functioning primarily as a storyteller, simply recounting events rather than discussing in detail his feelings and emotions: "her side was warm; her fawn lay there waiting" (10). With this approach, we tend to consider the speaker less and the actions and events more. The significance and feeling of what is actually happening comes to the forefront. This way, we, as readers, experience the scene first-hand rather than experiencing and reacting to the speaker's emotions first.
Instead, we are allowed to see and feel the scene for ourselves. By choosing this kind of steady, even-toned speaker, Stafford makes the action in the poem much more immediate and that makes the impact on the reader much greater. Good choice Bill.