Yep. I have insulated game bags (Trophy Bag Kooler) that I immediately put my deer into after making a kill. I pack the cavity with ice to get it cooled as quickly as possible.
I also get it cut and in the freezer asap.
Growing up, we always just threw our deer into the back of a pickup where they would sit for the entire weekend, and then they would hang in an unheated shop for a few days. We also had someone else do the processing, who most likely did gang processing. It was always gamy.
When I started doing my own, it wasn't gamy. I had gotten the kooler when I took up bow hunting since that season starts when it is warmer.
I've heard some conflicting reports on aging of venison. I've heard that venison lacks the proper compounds to age, and also that aging makes it better provided you keep it at a consistent temperature. Lacking the facilities to attempt aging on my own, I can't say for sure which is true. I do know that my venison that I immediately cool and process had been much better than the "barn Aged" venison we had growing up.