I have only ever been a part of 1 hunt where we left an animal over night, and it was only because we thought it might be the only way we would find it.
My brother shot an elk, but not very good. We tracked blood until dark, and just couldn't really follow the trail anymore. There wasn't enough blood to do it via headlamp. So we marked the last blood and left for the night, called a buddy with a blood tracking drahthaar, and went back at it in the morning.
(Side note: Let me say, those dogs are freaking amazing!) We found the bull the next morning, still alive, and my brother finished it off. So no worry of meat spoilage on that one...luckily.
I spoke to a well-known wild game butcher on the Wasatch Front about meat spoilage after killing elk. I had a cow tag I killed in mid-December when the temps where I was hunting were below zero, and the temps back on the Front were highs in the teens.
He told me if I can't get the gutted elk to him within 2-3 hours, I better skin it or the meat will start going bad. And he recommended skinning it anyway even if we could get it to him within 2 hours, even in those cold conditions, because you need to get meat cooled ASAP for the best results, and the hide simply will not allow the meat to cool.
So I think with a dead animal overnight, even in cold weather, the meat is probably toast. Which makes me never want to leave an animal overnight unless I absolutely had no other choice, like with my brother's bull a few years ago. We didn't really have another option under the circumstances.