We are lucky in the U.S. to have meat processed in the most controlled and sanitary system in the world. I have been through many of the major processors packing houses and they are all subject to the stringent federal inspection and H.A.C.C.P. requirements. ( H.A.C.C.P. = Hazard Analysis Critical Control Points. Originally developed by NASA for the space program)
When and animal is slaughtered in is immediately rinsed (up to three times at some packers). The primals (Beef & Pork) are then separated in <40 degree cutting areas and go into vacuum bags and are quick chilled, and into a cooler. From start to finish it is not much more than 30 minutes before it's vacuum packed and in the cooler. Considering that the animal is virtually bacteria free after slaughtering and it is put into a vacuum bag and chilled to 32 or less, (meat freezes at 28 so they can chill to less than 32.) it will keep for several weeks if kept at less than 34 and the vacuum bag is not opened.
Once the bag is opened then we in the supermarkets are locked into the 3 - 4 day rule with our products just like you are with the venison.
The difference with the venison is the exposure to bacteria all along the way after being shot. We gut them on the ground, drag them out through the woods/field, throw them in the trunk or pickup, hang them in a tree, shed, garage, etc. Then they're cut in the kitchen at 65-70 degrees. There is no way that we can control the exposure to bacteria in wild venison the way a H.A.C.C.P. certified packing plant can with domestic animals.
I challenge any of your locals to put venison trim into a refrigerator and then eat it after 2 -3 weeks! That's an invitation for food poisoning.
Dave