I've been a professional chef for 25 years and have never heard of this.
Getting it to the fridge quick is to cool down the meat so that bacteria does not begin to grow and spoil some of your meat.
Most wild game does not need to hang for any specific time as there is not the same ammount of connective tissue as in beef. So a day or two is plenty for hanging. I process my own venison and I find it much easier to work with if the meat is well cooled and firmed up.
As far as early season vs late season, you will find perhaps a difference in flavour mainly due to diet and fat content.
Hope this helps.