I've never prepped a deer that had been hit by a car before, so I can't comment on that part of it. Normally, I would gut it ASAP. Then, depending on the weather, I would be hurrying my but off to get the meat, or not so much so if it's cold.
My husband hangs them from a tree and completely skins them. He cuts the bottoms of the legs off with a hand saw when he gets that far along in the skinning. He then uses the saw to cut the back quarters off completely, and the shoulders. He takes the tenderloins and flank steaks off with a knife. All of this goes into big coolers filled with water, salt, and vinegar. At that point, it can be cut down from there at my leisure. It should soak in that brine mixture overnight. If it's warm outside, add some ice to the mix. Or bring it inside and cut it into the desired portions and soak it in bowls in the refrigerator overnight. It's a lot of meat, and a lot of work, so don't expect to get it all done quickly. And a knife sharpener will be your best friend.
All that is fine and good if you are a big strong man. That's not how I do it at all when I'm alone and someone drops a deer off. Which happens at least once a year, usually more.
If you want more details on the girly way, send me a PM! My husband calls it the coyote way, so you can maybe imagine what the carcass looks like when I'm done. It's all good though. However you do it.
Just for clarification, I think the backstrap and tenderloin are two different pieces, at least from what I was taught. The backstrap is along the back of the spine, from shoulders to hip, a fairly decent sized piece of meat. Then the tenderloin, is also located along the spine, but it's on the inside of the spine, between the spine and the body cavity.