I just retired as a health inspector for restaurants, etc. Using a clean plastic bag will prevent cross contamination between the meat and the cooler and actually protects both by preventing spread of bacteria between cooler and meat.
Its important to seal bag to keep out water from ice. Ideally, you would wash off the meat and place it in the bag and avoid contamination from unclean hands. Food science research shows that raw meat or cut vegetables require disposal if left above 41 degrees F for 4 hours or more. Remember that the venison is body temp when you start so hurry. Above 41F is danger zone , but between 70 and 100F is really bad--rapid bacterial growth. 135 or above is correct hot hold temp for items on food line--so danger zone goes from 42 to 134F.