I agree with "get it clean and cool as fast as possible". Not sure I buy the bleeding it out. If the heart has stopped beating its not going to bleed out.
Depends on whether you totaled the heart, or just wrecked the lungs, liver, etc. (though that also amounts to bleeding it out, just, internally).
The heart, much like most of the rest of the muscles, will continue to try to work, long past it's supposed expiry date, so the worst result of cutting the harvest's throat, is that it does not markedly improve anything, and at best, it may aid in clearing that much more blood from the system, which ain't a bad thing.
Based on several butcher's comments, and personal experience, it seemed to me that the guys that had the best experiences with their game harvest, were those that took the time and effort to keep the meat as clean as possible, process it to cooling, as fast as possible, and to transport it in as cool conditions as possible. I knew a couple folks that built portable 'coolers' using insulation foam panels and portable air conditioners, so they had a cool place to hang their game. I also knew a couple guys that took a freezer and genset with them on a sheep hunt. Sheep was cut and wrapped, and in the freezer, before they left camp. Head, horns, hide, and any other body parts that were REQUIRED for the registration process, were kept together, and they had no problems. They had great meat!
One butcher (Moose Jaw SK) told me outright, that the guys that thought they needed to hang a deer, usually ended up with half of it on the floor as trim/waste, as it was half way to being jerky already. I was at another trusted Butcher's place (North of Edmonton) dropping off an deer to be cut, and the Butcher outright refused to let the guys unload the moose carcass they had in their truck, because it was filthy, and he was not having that in his cooler...
Edited to add: Deer store their fat differently than beef. Beef are a lot like us, they store their fat inside the muscles, creating marbling. That and the enzymes spoken of earlier, allow beef to 'Age', and become more tender. Deer that are carrying a lot of fat, store it between the meat and the skin. I have had prairie White Tail does, that had fat layers that were inches thick, across their backs, and once the fat cooled down, you could separate the fat from the carcass with your fingers. Deer fat is not a palatable fat in general. Good enough when hot, but when cold, feels like a mouth full of gravel or sand, except sticky... Not great...
Never shot a buck that was carrying excess fat. Was told that a lot of them actually die off over the hard prairie winters, because they ran themselves bare chasing the girls...