This is season #40 for me. I've tried a bunch of different ways but for several years I had an extra fridge that we would put our boned-out meat into for 10-14 days.
I'm still not sure the meat tasted any better than after immediately butchering, but there were two unmistakable advantages. First, every cut was noticeably more tender. Second, the sinew was MUCH easier to strip/remove.
Those two reasons alone make aging worth it to me, if I am in a position to do so.
As an experiment, we hung a skinned, young doe in the garage one winter. Dad killed it in early-mid December. We did our best to keep the temp around 40° (not difficult in IL).
After a couple weeks we started cutting enough meat off the carcass for a meal. There was a small amount lost due to outside layer getting hard. However, we just removed that thin shell with a fillet knife like you would removing skin from a fish.
That doe lasted us into February and was the best venison I've ever eaten. It's not a practical way of doing it, but I hope to build a cooler one day to try it again.