I've always gutted them where they fall. Cutting the throat is a waste of time, they have already bled out internally. Throat cutting is for slaughtering hogs, to save the blood.
I've also always processed my own deer, antelope, bear, elk, and buffalo. With deer and antelope in particular, you can do a clean skinning and gutting job. However, when they get to the butcher, they are all cut and mixed with everyone else's meat. Then you get back SOME of your meat, with someone else's gut shot hairy carcass deer added in. Much better to learn to do it yourself.
A couple other things, washing out a carcass isn't a good thing. It promotes bacteria growth. a thin coat of blood coagulates on the body cavity, and helps protect the meat.
Meat saws are a bad thing on deer an antelope. The bone marrow in those will spoil the flavor of the meat. I'm not all that hot on using them on the larger animals either, and don't.