For approximately 30 years I leased and managed a 4600 acre ranch in West Texas, I sub-leased to hunters. I usually leased to a maximum of eight hunters each year, some maintained their lease for the entire time I had the lease. Now to your question.
I have seen deer killed with just about every caliber on the market. One hunter shot a deer 7 times with a .17, said they were all heart/lung shots, we ain't found him yet. In the .22 class bullets, I have seen a lot of deer killed with them, but the .22 is likely responsible for the loss of more deer than I can count. I can't count the number of times I spent half the night looking for a deer that had been "hit" with a .22 bullet.
The problem here is not the size of the bullet, it is the ability of the shooter. Oh, I have talked lots of hunters that were excellent shots at the range shooting from a bench. Shooting off-hand, or under field conditions they were not as good as they thought they were. Once exposed to a large buck, or a first time deer, the adrenaline starts pumping and no telling where the shot may go. I remember one case where we had two friend hunting together, one was to video tape the other when shooting a deer, now mine you, this ole boy was a very good shot at camp while zeroing the gun, he cut the X ring out, but when he took a shot at a large buck you could clearly see dust kick up five feet over the deer's back, a clean miss, yet again on the range he continued to cut the X ring out.
My point, it is not the caliber of the gun in most cases that is responsible for a clean kill, it is the ability of the shooter!