I've never fooled around with a deer's private parts, and I've never had tainted meat. I gut them quickly. Just get the guts out, string it up and let it drain and cool. If you have acess to cold water, wash out the cavity.
I wasn't taught that way. I was taught as a kid how to do it exactly right, like surgery. I found out 40 years ago, you don't need to be all that careful. The only thing that matters is to get the guts out ASAP, hang it up.
IMO, gutting sloppily or perfectly, the biggest mistake is heading for the vehicle too soon. A deer won't cool down in a vehicle like it would hanging, draining, and propped open with a stick.
But, I don't hunt where it's warm, so maybe there are better ways.
Two more points: learned these from experience. If you kill a deer that has been running hard, chased by dogs or whatever, the blood is hot throughout the body. Regardless of preecautions to cool it, it won't taste right.
The other thing: If you butcher in the woods like me, don't pile the meat in one pile for transport, like in a garbage bag. Spread it out so each piece can cool.